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Nonlinear Partial Differential Equations - Science topic

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During my master studies, we developed a numerical code to solve sine-Gordon equation in a class of discontinuous functions and I want to implement it on real problems. As I know, Josephson junctions and superconductivity give rise to this PDE.
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It arises for instance in Josephson junction arrays. You can see that in different papers and books. See for instance
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Dear colleagues, we know that getting a new research paper published can be a challenge for a new researcher. It is even more challenging when considering the risk of refusal that comes from submitting a new paper to a journal that is not the right fit. we can also mention that some journals require an article processing charge (APC) but also have a policy allowing them to waive fees on request at the discretion of the editor, howover we underline that we want to publish a new research paper without APC!
So, what do you suggest?
We are certainly grateful for your recommendations. Kind regards! ------------------------------------------------------------------------------
Abdelaziz Hellal Mohamed Boudiaf M'sila, University, Algeria.
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Cubo, a mathematical journal
Is very good for this fields of mathematics and more than it.
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what are the mostly used and applicable existence and uniqueness theorem for nonlinear partial order differential equations?
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There are some basic existence and uniqueness for PDE that can be explained from the Arzela Ascolis theorem
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I have to solve four nos. of equations with 5 nos. of unknown variables, two of these equations contains second order partial differentiation equation and two of them having first order partial differentiation equations. I don't know how to solve these equations and get the solutions of these variables ?
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I agree with Jose Risomar Sousa
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It is well confirmed that the nonlinear evolution equation is widely used as a leading mathematical equation for describing the physical significance of many physics branches. But it is a challenging task to solve them. Recently, many researchers have proposed various types of analytic methods to construct the soliton solutions. Unfortunately, most of these methods lead to equivalent solutions. In my opinion, the methods with the equivalent solutions of NLEEs are not helpful for further verification in the laboratory. That is why my discussion topic is "Which are the most appropriate analytic methods to construct the soliton solutions for nonlinear evolution equations?".
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It's ok, sir.
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Pleas see the attached file.
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In 2010, Dr. Khmelnik has found the suitable method of resolving of the Navier-Stokes equations and published his results in a book. In 2021, already the sixth edition of his book was released that is attached to this question for downloading. Here it is worce to mention that the Clay Mathematics Institute has included this problem of resolving of the Navier-Stokes equations in the list of seven important millennium problems. Why the Navier-Stokes equations are very important?
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OK, thank You, Christopher,
I hipe it can be done already to the end of this year but not to the end of this millennium.
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In mathematics and physics, a nonlinear partial differential equation is a partial differential equation with nonlinear terms. They describe many different physical systems. I'm especially seeking the stability approaches of fractional nonlinear partial differential equations that do not require linearization. Share your opinion.
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Azhar Iqbal
Thanks
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Dear Colleagues,
What are the importance of nonlinear partial differential equations in the development of Science and Technology?
For example a modern world. such as holographic touch screens, flying cars, robots doing homework and a hotel on the moon. so What do you think?
I value the insights and guidance you provide!
kind regards!
Hellal Abdelaziz
M'sila, Algeria University.
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Applications of nonlinear PDEs has a long list ranging from science to engineering.
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Hello, I want to learn how to solve a system of nonlinear elliptic partial defferential equations by using Matlab toolbox (pdetool), so, if there is a good starting point I should start with it first.
Thanks in advance
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This video could help you to get start with it
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- Could you please point me out to some Computer science, and Computer Engineering applications modeled, described, or analyzed using partial differential equations?
- Preferably, involving heat, reaction-diffusion, Poisson, or Wave equation.
- If possible in fuzzy environment.
Best regards
Sarmad.
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I suggest that you read
"Modeling Information Diffusion in Online Social Networks with Partial Differential Equations".
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I'd like to solve the following non-autonomous, non-linear first order differential equation, which is a result of (quite straightforward) chemical kinetics:
dy/dt = a*exp(-k*t) - b*y^2 - c*y with a,b,c,k > 0 and y(0) = 0.
Is there an algebraic solution? I tried all general methods including Laplace transforms, to no avail. WolframAlpha gives a solution using Bessel and Gamma functions, which is completely unstable in my region of interest.
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can we consider t at infinity to solve this equation?
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Hi, I hope you are keeping yourself in good health.
I am working in Matlab nonlinear parabolic model with finite element method for image processing. In effect, my idea is to turn images into mesh and find it nodes, elements, and edges, then use it in code which I have to solve nonlinear parabolic by FEM.
My question is what the code can I see results of this problem? Or did you have any idea or code Matlab about it?.
I want need a code Matlab about the finite element method for image processing.
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The method is quite extensive, I know that you can work with it in the pde toolbox (matlab). There is a book in russian with examples of matlab code. Perhaps this book will help you.
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I'm looking for Mathematics journals indexed in both Scopus and Clarivate at the same time, in the core of approximate-analytical solution of fuzzy partial differential equations. All help appreciated.
Best Regards
Sarmad.
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Hi, to all experts on the Finite Volume method.
Is it absolutely essential to linearize a non-linear source ,term, of a non-linear PDE during the process of solving a PDE with finite volume method (FVM)?
Few times in textbooks and online lectures, I have saw, experts tend to linearize the non-linear source terms( for example in a 2D Poisson equation. Lets say u"-exp(u)=0 ) during the process of the discretization of a PDE with finite volume method.
However, considering the approximation of the piece-wise constant , I expect the non-linear source will add non-linear terms to discretized equations (i.e., exp( u (x_p,y_p) ), where x_p and y_p are coordination of the center of the control volume) and one can use the Newton's method to solve a set of nonlinear algebraic equation.
For example, we could have a final set of the algebraic equation as Au - diag( exp(u) )=b, where "A" is the coefficient matrix, resulting of discretization and "b" is a constant vector containing the information of boundary conditions.
What harm it could have, on the conservative nature of FVM, if one processed similar to what I have explained on above example.
Many thanks of you, if you stop by and take look at my question. :-)
Vahid
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Dear Vahid,
The linearisation and the spatial discretisation are two independent aspects. To linearise or not is a decision you take regardless of the method you use, being FVM, or FEM, or DG, ...
If you have to solve a nonlinear elliptic PDE (as the one you mentioned), you have at least two options:
- Write down your linearisation scheme before discretisation; you are generating a sequence of linear elliptic equations, which you can discretise and so you end up with a sequence of linear algebraic systems.
- Discretise the given nonlinear elliptic PDE and generate in this way a nonlinear algebraic system of equations; to solve this you apply an algebraic nonlinear solver, which reduces often to solving a sequence of linear algebraic systems (as encountered e.g. when applying Newton's scheme).
In both cases, solving the linear algebraic systems generates the sequence of numerical approximations, which hopefully converges (oproving the convergence would be even nicer :))
Good luck,
Sorin
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we know that path planning for mobile robot is one of the most fundamental and complex problems in robotics. PDEs ( partial differential equations with variable exponents ) have been used in a variety of science areas, such as Mechanics, Calorific, Image processing, Image restoration, Electrorheological fluids and so on. Hence we want some references in these areas!
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let's express our thanks for your answers
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we know that Nonlinear Partial Differential Equations have a remarkable ability to predict the world around us. They are used in a wide variety of disciplines, from biology, economics, physics, chemistry and engineering. They can describe exponential growth and decay, the population growth of species or the change in investment return over time. Therefore we want some references in this field . Thanks
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let's express our thanks for your answers.
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A fundamental question for any PDE is the existence and uniqueness of a solution for given boundary conditions. For nonlinear equations these questions are in general very hard, hence we need an answer for the above question . Thank you very much!
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The answer is a little long and complicated, I recommend you read about
entropy solution, normalized solution and weak solution for a nonlinear parabolic equation in a paper by clicking the following link:
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Yes, there is a new method which is called Piecewise Analytic Method (PAM). It does more than Runge-Kutta.
1. PAM gives a general analytic formula that can be used in differentiation and integration.
2. PAM can solve highly non-linear differential equation.
3. The accuracy and error can be controlled according to our needs very easily.
4. PAM can solve problems which other famous techniques can’t solve.
5. In some cases, PAM gives the exact solution.
6. ....
You can see :
Also, You can write your comments and follow the update of PAM in the discussion
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I stopped working in this field nearly 20 years ago. But even than there existed several methods that outperformed RK4 by lengths, the comparison is done on the estimated accuracy obtained by the same number of calls to the ODE's right hand side function. Have a look in the following books
Hairer, E. and Nørsett, S. P. and Wanner, G.: Solving Ordinary Differential Equations, Part I - Nonstiff Problems, Springer Verlag, 1987.
Hairer, E. and Wanner, G.: Solving Ordinary Differential Equations, Part II - Stiff and Differential-Algebraic Problems, Springer Verlag, 1991.
There the methods (for non-stiff ODE) of Dormand/Prince are highly recommended (I used one in my time). Somewhere in the middle of the first book is a twosided graphics where the orbits of a specific problem are shown obtained by different methods (and the same maximum number of rhs calls). The orbits of the chosen ODE are known from theory to return to the starting point. The impressive fact of the graphics is to show how good this feature is reproduced by the methods applied: the orbit of 1-step Euler metod leaves the pages and does not return, the orbit of the RK4 method leaves a large gap between start and end points, the orbit of the DP method without stepsize control leaves a small gap, and the orbit of DP with stepsize control leaves no visible gap; even more, the last method needed much less rhs calls!
This was state of the art 20 years ago. Maybe that meanwhile better methods have been developed, maybe the initially discussed PAM is one. Anyway, I recommend that you replace the RK4 method in your code at least by the DP4/5 method. If you use Mathlab then apply ode45 as ODE solver, if your code is in Fortran or C then search the internet for DOPRI5.
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Dear scholars,
I am working on finding a numerical solution for an equation set with implicit boundary conditions on Matlab. The equations are expressing a nano liquid flow problem which are derived based on including continuity, momentum and heat transfer equations. The equation set and the related boundary conditions are attached. Except f and θ which are the functions of η, all the parameters are the properties of the fluid and known.
So, could you please tell me which numerical method is proper to treat this problem?
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Looks like some sort of convective boundary layer problem from a cylinder.
These equations and boundary conditions form a typical voundary value problem. The equations only look complicated because of a large number of constants due to the nanofluid presence. If you are able to solve the Blasius boundary layer equations using a shooting method, or, better, the boundary layer equations for convection from a vertical surface - these date back to the 1950s, then this is just more of the same with no extra difficulties.
Re the boundary condition at infinity, a sufficiently large eta_max will do, but one will have to increase that gradually in order to check that your chosen eta_max is large enough. But I have a feeling that the solutions for theta and f' may well decay algebraically (a negative power of eta) rather than exponentially, so the "compactified" rescaling may eventually need to be done. But I would try without it first in order to make sure that your code runs and gives physically realistic looking solutions.
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I am trying to solve Coupled Partial non linear differential equation using pseudo spectral method. The problem is time, Its taking lot of time to find the roots even with few numbers of collocation points. Also how to ensure that the roots are real.
The program I am using is MATHEMATICA.
Any suggestion will be helpful. Thank You.
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Following
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I need a detailed step by step explanation and workings of how to derive the recursion relation as contained in the solution I enclosed in a box in the attached solution by W. Hereman et al.
I need to apply the method to other nonlinear PDEs.
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I have tried this with equations that were nonlinear by containing a quadratic term. I was never able to solve the recursion relations but I was able to derive them. The trick, for quadratic terms, is the following. Suppose that each expression in the equation is expressed in powers of x, but a quadratic term in the equation has the double sum
Sum from m = 0 to infinity of sum from n = 0 to infinity of a(m,n) x^(m+n)
The trick is to express this as
Sum from n = 0 to infinity of (sum from m = 0 to n of a(m,n-m)) x^n
By doing this, the outer sum contains only a single power of x (the nth power). The inner sum can be equated to the coefficient of x^n on the other side of the equation. This produces a recursion relation for the coefficients but the next trick (solving them) is something that I can't help with.
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Hello,
I have a strange bug in my code involving the use of FFTW3. I already wrote the qeustion on Stackoverflow: https://stackoverflow.com/questions/53518451/why-is-fft-of-ab-different-from-ffta-fftb
Please, if anybody have some time to spare, could you have a look?
Thank you very much!
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Finally I solved the problem! If anybody is still interested, you can have a look on Stackoverflow, I posted an answer.
PS: Hussein M. H. Al-Rikabi Fatima Faydhe al-Azzawi I insist, the Discrete Fourier Transform is a linear operation.
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I have a set of coupled ODEs:
J = mu*e*n(x)*E(x) + mu*K*T0* dn(x)/dx
and
dE(x)/dx = (e/eps)*[N_D(x) - n(x)]
These are the drift diffusion equation and Gauss's law for a unipolar N+ N N+ device. The doping profile N_D(x), the mobility mu, T0 are known. The DC current, J, is also known, as are the boundary values: n(0) = N+, N(L) = N+. I want to self consistently solve the above two equations using finite differences, but I am unsure how to go about doing so.
Thanks in advance!
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Hello,
I have written simple 1D drift-diffusion solver examples using finite difference and Scharfetter-Gummel discretization which you can find here:
I'm happy to answer any questions.
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What is the explanation using pde equations to demonstrate coupling 2 ode equations and spatiotemporal behavior?
or what is the exact meaning of using pde in ode equations? does it related to the location of the variables which can affect the values of the other variable in time?
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This is a worthwhile question.
The relationship between PDEs and ODEs is given in detail in a bit hefty 2018 paper:
Finite PDEs and finite ODEs are isomorphic
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Hi guys, I am currently building a mathematical model for simulating cable motions underwater. It involved a partial differential equation.
However, I am not really familiar with using Matlab. Is there have any tutorial or examples you can suggest? I will appreciate if you could give any advice.
Thank you for your time.
Ps. It is attached the partial differential equation and related article.
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Keep in mind that there are few cases in which you can calculate exact solutions of initial and boundary value problems. In most cases we are forced to calculate approximate solutions in discrete domains, that is, with a finite number of points and not in the continuous domain, which requires replacing the problem in PDE with a problem expressed through relationships based on finite differences, finite elements, finite volumes and others. You can use Newton Raphson's method to solve this approximate system in the finite set of points. Not the original initial and boundary value problem .
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best (achievable) x-reg for singular solutions of nd incompressible euler equations by Székelyhidi jr.+de lellis and remarks/links with pressureless (regular) sols ?
--on solutions "convex optimized" by l. Székelyhidi jr. and c de lellis (and descendants) of nd (or 2d, 3d if there is a difference for LS-CD frames and if so, then how ?) incompressible euler equations, could anyone tell what is the best x-reg (so better than C ^ s, s about 1/3) that LSjr+CD or their descendants, 1 / have already, 2 / could be expected and what are the obstacles getting (or not) a better x-reg ? (all with the same question in t-reg and mixed (x, t) -regs), this regardless of their first motivations, that is, breaking the uniqueness (more generally realistic up to C ^ s for all s <1) and onsager (related to s=1/3) or the final answer is close to s=1/3 (and then why)?
--Can anyone confirm that the solutions coming from the classical and regular theory (that is to say not LS&CD theory), which are the particular solutions that are  pressureless (see in 2 / 3d, majda diperna pl lions but also in Rn, all n, in my jmpa95 and thesis92-ch3/90 etc, quoted by yudovich) are out of reach by LSjr+CD theory and their singular frameworks (because for LSjr+CD, the pressure is, at first, basically + - equal to cst lul ^ 2 and then p = 0 gives u = 0 or all of this can be (partly) overcome and how, for example, include these sols in LS&CD setting?) = What could be said about these sols regarding the LS&CD theory ?
--regular theory =eg, for a (short) interval of non nul times, +- Du or rotu (:Du-tDu) are in x, C^s (s in ]0,1[), C^o, or L°° or bmo or (eg in 2d:) Du or rotu in Lp(loc), for one p in [1,oo] etc
--LS&CD singular theory at the state (?)= eg , for a (short) interval of non nul times, u itself (and not its gradient Du) is only in C^s, s in ]0,1[ or s<1/3 or s= 1/3 etc and eg nothing (yet?) on (atleast L1(loc)/loc-measures) DERIVATIVES of u, eg Du or rot(u) obtained (and conserved) for non nul times (even if it is supposed at t=0) etc
-- Are there (zones of) junctions or intersections (and where and what) between regular theory and LS&CD singular theory, existing same common sols (even particular) for these 2 theories ?
--eg 2d, or 3daxi, rotu (and/or its moments 0,1,2 and/or the axi structure) could be? (even "half" or "under" or partly in some way) conserved in SDth as in regTh etc ?.
--questions extended of course to all flu mech models already treated (by extensions from IncEE) by SDth. 6/7/18
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on the comment of P. K. Karmakar = the sols and theory by
Székelyhidi jr+de lellis on, at first, the incEE (and afler, other models of flu mech), date from c2007 (excepted mistake of my part for all), not before, and it is of a hight level of theorical maths (a talk at the bourbaki seminar was decided and made on it) one of the most important results on incEE for the last eg 20 years. the book you said is earlier, ie 2002
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- Could anyone please help me to some Biochemistry, Genetics and Molecular Biology modeled, described, or analyzed using partial differential equations? "the model is very appriciated"
- Preferably, involving heat, reaction-diffusion, Poisson, or Wave equation, an If possible in fuzzy environment.
Best regards
Sarmad.
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Turing patterns in reaction diffusion equations.
SIR modelling
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- Could you please point me out to some successful Medical sciences applications using partial differential equations?
- Preferably, involving heat, reaction-diffusion, Poisson, or Wave equation.
- If possible in fuzzy environment.
Best regards
Sarmad.
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Dear Professor, In mathematical modelling for drug delivery we are using PDE.
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Could you please point me out to some successful Signal, image, or video processing real life applications using partial differential equation?
Preferably, involving heat, reaction-diffusion, Poisson, or Wave equation.
If possible in fuzzy environment.
Best regards
Sarmad.
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Many nonlinear equations are solved for soliton solutions using Hirota bilinearization method but those equations are proven to be integrable either in inverse scattering sense or Lax pair method. Is it correct to use the Hirota bilinearization for nonintegrable systems? 
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A short answer is: yes, it is, because the existence of HIrota bilinear representation per se does not guarantee integrability. It more or less guarantees existence of a solution of a special form that in many cases can be interpreted as a two-soliton solution. Integrability, however, requires existence of four-soliton solution, see e.g.
Hietarinta, J. Hirota's bilinear method and its connection with integrability. Integrability, 279–314, Lecture Notes in Phys., 767, Springer, Berlin, 2009.
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In reaction-diffusion systems, numerical simulation in two dimensions is an important to compare the results. I need some suggestions where I can start. Especially, I'm interested in Python programming. Which papers, books or anything else will be helpful for me to study the reaction-diffusion-advection systems.
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Is there any program capable of simulating partial differential equations formulated in curvilinear coordinates, which uses these coordinates internally for the computations (i.e., no transform to cartesian coordinates)? And is there any program doing bifurcation analyses for such equations? And, finally, is there any program which can do both?
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Dear Mario, I use Mathematica and for solving pde in curvilinear coordinates (e.g. spherical or cylindrical) first I transform equation from cartesian to curvilinear coordinates, then I (hopefully) solve it using Mathematica internal built-in functions like NDSolve; I think that these internal functions apply numerical methods directly to curvilinear coordinates. See link below for an example from Quantum Mechanics. Gianluca
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Is BVPh Mathematica package valid for linear operator with respect to "t" derivative in one dimensional PDEs by Homotopy analysis method?
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I thank that BVPh Mathematica package based on the Homotopy analysis method is well used just for non-linear ODEs and PDEs. 
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Hi, I resolved a classic advection diffusion equation. The initial and boundary condition is C(x,0)=0; C(0,t)=C0; dC/dx (x=inf)=0. In fact, for x=0, this is a standard first type boundary condition. However, the reviewer asked me why not to use the third type (D(dC/dx)+vC=vC0 for x=0) for the mass conservation. I'd like to know which of the two types is often used, the third type is better than the first type? Thank you
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Hello Enze,
The third boundary condition you are asking for is Robin boundary condition. It is like mix of dirichlet and neumann boundary condition (a linear combination). This kind of boundary condition is normaly used to model thermal problems. You may be thinking what is the reason of this kind, what is the applicability. The most common usage of it is when it is necessary to model  convective and diffusive fluxes simultaneously which is quite common for solving thermal problems.
You may be thinking: "Is there any outher applicability?". Yes, nowadays I am using this kind of boundary condition to model lymphatic vessels, as you can read at this article https://www.researchgate.net/publication/310425146_Interstitial_Pressure_Dynamics_Due_to_Bacterial_Infection.
Sincerely
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The following two equations are Richard's equation, but with different signs in the K (hydraulic conductivity) term. What makes the difference? Why it is so ? Most of the literature used Richard's equation with negative sign in K term.
Reference for the equations:
1. The equation with positive sign is from Wikipedia
2. Another equation with negative sign is from
Numerical solution for one-dimensional Richards’ equation using differential
quadrature method
by Jamshid Nikzad, Seyed Saeid Eslamian, Mostafa Soleymannejad, and Amir Karimpour
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Suppose you introduce a new independent variable u=-z.   Then your negative sign system becomes the positive sign system.  The difference is just a matter of choosing the spatial coordinate direction; there is no discrepancy.
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I have 2 nonlinear parabolic PDEs which are coupled (heat and mass diffusion equation). I am looking for a method with good convergence and stability. The language I should write my code in is FORTRAN. 
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A couple of ODE systems is obtained by the method of lines (Finite Difference in terms of space) from a PDE, e.g., u_t + v*u_x = 0 is transformed into u(i)_t = -v*(u(i+1) - u(i - 1))/2h or -v*(u(i) - u(i-1))/h. However, the divergence of the discretized ODE systems is totally different between these two discretization schemes,  and they are zero and -v/h respectively, leading to a divergence-free and nondivergence-free ODE system. So, what happen here?
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A couple of points:
1.  The 1/h scaling is just the result of approximating a derivative (u_x) by a divided difference, so is perfectly appropriate: one must only interpret the formula correctly.
2.  The first order system u_t + v*u_x = 0 is more easily treated by the Method of Characteristics (noting that u_t + v*u_x is a directional derivative in t,x space).
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I would like to derive the Navier Stokes Equation from three first order ordinary differential equations (shown in the attachment). I would be glad to have your expert opinions and suggestions. 
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It isn't possible. The Navier-Stokes equations are a set of coupled, nonlinear partial differential equations. Your quoted equations are uncoupled, linear ordinary differential equations. It may be that your system has been derived from the NS equations, but this would have been subject to all sorts of simplifying assumptions for a special type of flow. Therefore, at best, your equations form a very small subset of the NS equations, and the latter cannot be reconstructed from the former.
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I am solving PDE heat equation using Galerkin FEM and it includes function of temperature in its parameters as exponential terms. I need solution of Temperature variable in terms of distance and time and it requires matrix formulation. This makes my ODE very complex and I am unable to discretize it.
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Could you write down the equation here please, so you can get the best help.
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I have attached a paper below which describes a multi-region method for solving for the wave characteristics around a submerged porous structure. I am struggling to solve the last four equations, (23) - (26), (32) and (33). My equations are indeterminate because of the starting points A(M1+1, q) and B(0, q). If anyone could please give me some insight into how to solve these four equations I would be very appreciative.
Thanks for your time.
Kind Regards,
Alex Wylie.
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The ``starting points''  aren't given-their values are to be determined by solving the system of equations.
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Hello,
Anyone could explain the physical meaning of the partial differential equation coefficients in the propagation of elastic waves in solid media?
Equations for isotropic solid media in 2D:
Equation 1: ρ (∂ ^ 2 u_x) / (∂t ^ 2) = (2G + λ) (∂ ^ 2 u_x) / (∂x ^ 2) + G (∂ ^ 2 u_x) / (∂y ^ 2) + (G + λ) (∂ ^ 2 u_y) / ∂x∂y
Equation 2: ρ (∂ ^ 2 u_y) / (∂t ^ 2) = G (∂ ^ 2 u_y) / (∂x ^ 2) + (2G + λ) (∂ ^ 2 u_y) / (∂y ^ 2) + (G + λ) (∂ ^ 2 u_x) / ∂x∂y
Recalling that I understand what the physical parameters (G; λ) themselves represent for the specific physical problem, but I do not know what is the physical implication of its use as a multiplier of each partial derivative.
Thanks in advance for the help.
Renato
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Ok Mr. Baeker, you helped me with this matter.
Thank you!
Regards.
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I am trying to derive weak-field Schwarzschild metric using Linearized Einstein's field equations of gravity:
[]hμν – 1/2 ημν []h = -16πG/ c4 Tμν
For static, spherically symmetrical case, the Energy- momentum tensor:
Tμν = diag { ρc2 , 0, 0, 0 }
Corresponding metric perturbations for static ortho-normal coordinates:
hμν = diag { htt , hxx , hyy , hzz }
With one index rised using flat space-time Minquoskwi metric ημν= { -1 , 1, 1, 1 }:
hμν = diag { -htt , hxx , hyy , hzz }
Trace of the metric:
h = hγγ = - htt + hxx + hyy + hzz
The four equations:
1) []htt – 1/2 ηtt []h = -16πG/ c4 Ttt
=> []htt + 1/2 []( - htt + hxx + hyy + hzz )= -16πGρ/ c2
=> 1/2 []( htt + hxx + hyy + hzz )= -16πGρ/ c2
2) []hxx – 1/2 ηxx []h = -16πG/ c4 Txx
=> []hxx - 1/2 []( - htt + hxx + hyy + hzz )= 0
=> 1/2 []( htt + hxx - hyy - hzz )= 0
Similarly:
3) 1/2 []( htt - hxx + hyy - hzz )= 0
4) 1/2 []( htt - hxx - hyy + hzz )= 0
Adding equations 2), 3) & 4) to 1) respectively, yield:
[]( htt + hxx ) = []( htt + hyy ) = []( htt + hzz )= -16πGρ/ c2
Solving the equations using:
[] ≈ ▼2 ≈ 1/R2 d/ dR ( R2 d/ dR ) for static spherically symmetric case; we get:
( htt + hxx ) = ( htt + hyy ) = ( htt + hzz )= -8πGρR2 / 3c2 – K1/ R + K2
Similar solutions for vacuum case, with Tμν= 0 would be:
( htt + hxx ) = ( htt + hyy ) = ( htt + hzz )= – K'1/ R + K'2
For the metric to be asymptotically flat:
K2 = K'2 = 0
For the solutions to be continuous at boundary, R= r, the radius of spherically symmetric matter:
- 8πGρr2 / 3c2 ≈ - 2Gm/ rc2
The remaining two constants must be:
K1 = 0 & K'1 = 2Gm/ Rc2
Therefore, my solution comes:
( htt + hxx ) = ( htt + hyy ) = ( htt + hzz )= - 2Gm/ Rc2
But, as per the literature, the weak field Schwarzschild metric must come out to be:
htt = hxx = hyy = hzz = 2Gm/ Rc2
Thus the solutions must come out to be:
( htt + hxx ) = ( htt + hyy ) = ( htt + hzz )= + 4Gm/ Rc2
I am not able to make out where I am making mistake. Can anybody please help?
Thanks.
Nikhil
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The Schwarzschild metric is the exact, spherically symmetric and static, solution of Einstein's equations, in vacuum, where the only parameter is the mass of the object-so the energy-momentum tensor of matter vanishes.  You won't get a singularity-and, therefore, a horizon-with linear equations. The horizon is a *global* property of spacetime (as separating the observers that will not reach the singularity in finite proper time from those that will). 
Cf. https://arxiv.org/abs/gr-qc/9712019, p.164 and following. 
One can, also, show-but it's harder-that it's stable under certain perturbations:  http://www.ctc.cam.ac.uk/activities/adsgrav2014/Slides/Slides_Holzegel.pdf
If you do want to study perturbations about a Schwarzschild metric, just write g = g_S + h and keep the linear terms in h. 
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Depending on the eigenvalues (and in fact eigenvectors) of A and B, not all entries of A are equally important in evolving the system states of (A,B). In fact, it seems that some entries in A, e.g. a_(ij), with some proper mapping could be zero because the effect of x_j on x_i is quite negligible as time proceeds. The problem is for a stable (or marginally stable) A how that mapping is defined to maximize the number of zero entries in A_s. I am interested in knowing all related results for this problem. 
I do not care if you know of some results on continuous-time systems.
Thanks in advance.
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You are 100% right however I am not interested in diagonalization. 
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According to Prof. Luca Bottura (in paper: Modelling stability in superconducting cables), the PDEs that describe the flow condition of helium in a single channel can be written as the form in the image I uploaded. However, the non-homogeneous terms bring some confusion to me. Taking the temperature equation for example, if there is no gradient for temperature and velocity along x-axis meanwhile assume there is no heat exchange between helium and other components, then because the first term on the right is always positive (so long as vh>0), the temperature of helium will keep rising. I wonder how to understand this .
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Dear Rui Kang,
May this paper help you,
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It's known that we can approximate a solution of parabolic equations by replacing the equations with a finite difference equation. Namely, we can solve parabolic equations by Difference Equation Replacement or Crank-Nicolson methods. I need Matlab code of Crank-Nicolson method for attached problem
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In  Matlab, package, search for,  Crank-Nicolson, there are some examples, which can help you.
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My problem concerns proving the existence of solutions to a set of linear PDEs on a semi-infinite strip. We have 2 variables, r which takes the range (h, infinity), h > 0, and xi which takes the range [-1, 1].
The issue is that I wish to solve the Cauchy problem near the boundary r = h, but on r = h the PDEs change type: here they are parabolic, and for r > h, they are elliptic. This is causing the problem that if I try to put the equations into canonical form, the variable that r is transformed to becomes singular on r = h.
What can be done here? The way it seems, if I use r = h for the Cauchy curve, and hence have a parabolic equation on the boundary, how can I define a solution arbitrarily near r = h if suddenly the type changes, for *any* r > h? Or am I thinking about it wrongly?
Thanks in advance for any pointers.
(EXTRA INFO: The equation in w(r, xi) is of the form
r^2*mu*(d^2 w/dr^2) + r^2*(d mu/dr)*(dw/dr) = f(xi)*(d^2 w/d xi^2)
with f(xi) >0 for all xi in (-1 ,1), and where mu(r) is a monotonically increasing one-to-one function such that mu(h) = 0 and where all d's are partial (except the one attached to mu). Transforming this to canonical form, we find we must use a variable like
alpha(r) = integral (from R = h to r) of [1/(R*sqrt(mu(R))) ] dR,
which clearly has problems with the lower limit - unless r = h in which case alpha(h) = 0, another thing which is slightly baffling me!)
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 Hi Erik. Your substitution making the term next to u_{rr} equal 1. Similar simplification one can make by dividing the equation over r^2\mu. Using the substitution mention by me, one can get rid of the first order terms. But which substitution to use depends on what method you are using to prove the existence. So it good to try them all and see which one works better. By exact solution I mean representation by integrals, which could be helpful if you prove the existence using iterations. I think power series expansion could work better, but under some analyticity conditions and restriction on class of solutions that are interesting from the physical point of view.  
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Solving PDE in Space domain only
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Perhaps a good initial approach starts by asking about the modeling:  Why do you believe (on physical grounds? on mathematical grounds?) that this coupled system actually has a solution?  If one is working with a perturbation of a smaller system, one might think of an iterative or continuation approach.   The numerics would follow the rationale...
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In relativistic mean field theory the Dirac equation becomes a system of first order differential equations. They must be solved numerically. I have problem in solving them.
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If you are using R  you can use the packages
and  
that include many robust solvers for stiff ODEs, in particular it includes all the FORTRAN solvers availabe in the  Test Set for  IVP solvers
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The inhomogeneous heat equation is described as follows: 
Ut - kUxx = f(x,t). In order to solve this inhomogeneous PDE, Are there any restrictions on the type of the function f(x,t)? Even / Odd or neither?
Thanks
Kulthoum 
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Dear Kulthoum Ismail,
Your PDE is linear and Non-homogeneous.The non- homogeneous function f(x,t) must  be  known real valued continuous function on given domain.It may be any function.There are so many methods for solving this equation under suitable initial and/ or boundary conditions.Some of them are:
I) Method of Separation (Oldest and simplest method)
II) Method of integral Transform
III) Adomian Decomposition method
IV) Perturbation Method
But the non- homogeneous function f  is nonlinear  in U then we have to put certain conditions on f. It is different issue.
Best luck!!!
DNYAN
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Background: I am an ok coder but have not learned to use either of the above programs, both of which are freely available to me at my workplace and both of which I have been told will basically do the job with general numerical methods.
Extra background: The models I'm working with are black holes, so they are singular at at least one boundary (the event horizon), meaning that an adaptive step size method must be used; and given that there are essentially more free parameters to a general solution at infinity than at the event horizon, we sometimes need to use a "shooting" technique where we take 'initial' conditions at (near) both boundaries and try to match them up in the middle.
I can work out/look up the numerical recipes for the methods - but any advice on which programming language is more suitable would be greatly appreciated. Also, show your working ;)
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C ++ is faster , but working with MATLAB is easier. If you have a very large data, that would be better to choose C , however if the data is not that big, MATLAB is better.
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To solve a given heat equation on the half line we can use the reflection method where the initial data is an odd extension (Dirichlet boundary conditions) /even extension (Neumann BCs). why do we choose an odd or even extension, is there any clear reasons for that? 
The method is impractical and cannot be generalized but it is a good way to understand the physical meaning of a given PDE, therefore any clear explanation would be appreciated!
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Hello!
It is connected with the fact that in the case of odd continuation, the corresponding solution will satisfy homogeneous Dirichler boundary conditions, and in the case of even continuation it will satisfy Neumann boundary condition. You can find proofs of these facts in 'Equations of Mathematical Physics' by A.N. Tikhonov, A.A. Samarskii, pages 254-257.
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E. H Doha
M. A. Abdelkawy
A. H. Bhrawy, or any other who knows the answer
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If you use explicit Euler method, the system can be solved very easily. Each equation of the system can be solved independently, similarly to an ordinary differential equation.  Things are different in the case of the implicit Euler scheme, since we need to solve a nonlinear system.  Maybe the answer will be "more accurate" if you give more details about your system.
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Hi,
 I want to solve a simple 2D steady rectangular domain in both analytical and Finite element solution and compare the error between them. But for this, I need to know the analytical solution at first, that is I need to take a function and then derive the boundary conditions along the domain ( Inverse heat transfer problem). I want to know what kind of function can I make an assumption so that it satisfies both Laplace equations and the boundary conditions ( Dirichlet and Neumann). I want to do it for steady state. Also, if possible explain me the procedure of how to get the final analytical solution.
Thanks!
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@ Brian
 Hi Brian,
    I am still skeptical about the solution to the question which I sent here few days back. I want to know who to find the source or the conductivity term in 2D heat when the analytical solution is known. And how to find that analytical solution???
What your response with respect to Poisson equation was correct, But I don't know how to find the conductivity or the heat source term from it. Please help me out with respect to this.
Thanks and Regards,
Sunag R A.
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I am confronted with a research problem in a field which is unfamiliar to me, and though I am a quick learner, I need to know where to start!
My problem concerns proving the existence of global solutions to a set of coupled integro-differential and partial differential equations. I do not need to find specific solutions. I have some scant experience with finite difference methods but I fear they will not cut it in this case, or at least they are slightly less-than-satisfactory here.
Can anyone please give me a) any texts/links they would recommend for this purpose, and/or b) any particular methods or theorems I should be googling?
Thanks in advance.
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An excellent book is:
Partial Differential Equations by Lawrence C. Evans, Graduate Studies in Mathematics, Volume 19, American Mathematical Society, Rhode Island (1998). This is a very readable textbook, that is also highly estimated by some of my colleagues in mathematics. Warning: Results on the existence of global (not only local) solutions are meagre.
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Hello please I have a question about Lemma 3 of the article "Uniquenss of Positive Solutions of Semilinear Equations in R^n" L. A. Peletier and J. Serrin 1983. On page 185 of the article there is "since u '(0) = 0 there exists a first point r0 where u' (r0) = 0" I want a detailed demonstration of this result because I see that " It has another case in other words, we can have u '(r)> 0 in the neighborhood of 0 why it eliminated this case and it only deals with the case where u' (r) <0 in the neighborhood of 0 (see Fig.1 of article) .and thank you very much my great teacher.
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salut envoie moi le papier et prisé bien votre question 
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In my research I faced with the problem of finding the solution if set of nonlinear matrix equations. I have found several articles devoted to specific form of nonlinear matrix equation, namely AX + XB + XFX + C = 0, (A,B,C,F - coefficient matrices, X - variables matrix) but I failed to include my problem to the form.
I'm interested in the equation, which can be describe as follows:
CT = Q and AT + BF + TS = 0, where A(nx x nx),B(nx x nu),C(ny x nx) and T(nx x (nx+np)) - variables matrices, Q(ny x np), F(nu x (nx+np)), S(nx+xp x nx+np) - known coefficient matrices. There is no doubt the problem has the solution, in general, solutions set are infinity, the goal is to find at least one of them. 
I tried to use minimization problem, but works well for specific matrix dimensions, namely, I can find the solution when nx<np.
Can anybody recommend me a book devoted to nonlinear matrix equations or better a set of matrix equations? Or is there some ideas about what I should to do to find solution?
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Ok, now I see it! Indeed, I did't think about the problem form this point of view. I will try to find something in this direction. Thank's a lot!
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We are interested in computing the largest Lyapunov exponent for a variant of the the Kurimoto-Sivashinsky equation in 1+1 dimensions.  Any suggestions?
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Dear Prof. Bradley,
 I suggest you check out the links below :
I hope this should help.
Best regards.
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A non linear system have 8 inputs and 2 outputs. I have parametric data set related to input output. What will be the best modelling technique?
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You can use  System Identification methods.
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I have a function of several variables like, f(x1,x2,...,xn). I have poroved that, the function is optimized at (a1,a2,...,an) by satisfuyng (i) firts order partial derivative exist and f'(a1,a2,...,an) = 0 and (ii) corresponding hessian matrix at the points (a1,...an)  is positive definite or negative definite for minimum and maximum, respectively.
Now theoretically, how can i prove that, the function f(x1,x2,...xn) is optimum at given points (a1,a2...,an)? 
Advance thanks for suggestion and give references for that. 
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Unfortunately, for a function defined on the noncompact domain $\mathbb R^n$ there is no way to derive a global statement from a local statement.  You would really have to use more information about your function, in order to derive the relevant inequalities.
Even in the simple case that the function has only one critical point, which is a local minimum, it is not true that this has to be the global minimum (e.g. $x^2(1+y)^3+y^2$, the example is taken from Fuchs & Tabachnikov: Mathematical Omnibus.)
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Hello,
I look for a full tutorial for the numerical solution of a Nonlinear PDE system by Matlab, Mathematica or Maple.
An example will be welcome.
Thank you
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And another tutorial for a single nonlinear PDE, with more details: http://fvt.simulkade.com/posts/2015-04-06-solving-nonlinear-pdes-with-fvm.html
I linearize the PDE and solve it in a finite volume tool I have written in Matlab: http://www.mathworks.com/matlabcentral/fileexchange/46637-a-simple-finite-volume-solver-for-matlab
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My function values are going negative while i need to restrict it to zero. How do i do that?(optimal control problem)
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Dear Sinha
Some of my papers
1- A. R. Seadawy, Fractional solitary wave solutions of the
nonlinear higher-order extended KdV equation in a stratified
shear flow: Part I, Comp. and Math. Appl. \textbf{70} (2015)
345–352.
2- Khater, A. H., Callebaut D. K., Malfliet, W. and Seadawy A. R., Nonlinear Dispersive
Rayleigh-Taylor Instabilities in Magnetohydro-dynamic Flows,
Physica Scripta, 64 (2001) 533-547.
3- Khater, A. H., Callebaut D.
K. and Seadawy A. R., "Nonlinear Dispersive Kelvin-Helmholtz
Instabilities in Magnetohydrodynamic Flows" Physica Scripta, 67
(2003) 340-349.
4- A.R. Seadawy, Exact solutions of a two-dimensional
nonlinear Schrodinger equation, Appl. Math. Lett. 25 (2012) 687.
5- A.R. Seadawy, Stability analysis for Zakharov-Kuznetsov equation of weakly
nonlinear ion-acoustic waves in a plasma, Computers and
Mathematics with Applications 67 (2014) 172-180.
6- A.R. Seadawy, Stability analysis for two-dimensional ion-acoustic waves in
quantum plasmas, PHYSICS OF PLASMAS 21 (2014) 052107.
7- Helal, M. A. and Seadawy A. R., Variational method for the
derivative nonlinear Schrodinger equation with computational
applications, Physica Scripta, 80, (2009) 350-360.
8- Helal, M. A. and Seadawy A. R., Exact
soliton solutions of an D-dimensional nonlinear Schrodinger
equation with damping and diffusive terms, Z. Angew. Math. Phys.
(ZAMP) 62 (2011), 839-847.
9- Khater, A. H., Callebaut D. K. and Seadawy A. R., General
soliton solutions of an n-dimensional Complex Ginzburg-Landau
equation, Physica Scripta, Vol. 62 (2000) 353-357.
10- Khater, A. H., Helal M. A. and Seadawy A. R.,
General soliton solutions of n-dimensional nonlinear Schrodinger
equation" IL Nuovo Cimento 115B, (2000) 1303-1312.
11- Khater, A. H., Callebaut D. K., Helal, M. A. and Seadawy A.
R., Variational Method for the Nonlinear Dynamics of an Elliptic
Magnetic Stagnation Line, The European Physical Journal D, 39,
(2006) 237-245.
12- Khater, A. H., Callebaut D. K.,
Helal, M. A. and Seadawy A. R., General Soliton Solutions for
Nonlinear Dispersive Waves in Convective Type Instabilities,
Physica Scripta, 74, (2006) 384.
13- Seadawy A. R., Three-dimensional nonlinear modified Zakharov–Kuznetsov
equation of ion-acoustic waves in a magnetized plasma, Computers
and Mathematics with Applications 71 (2016) 201-212.
14- M.A. Helal and A.R. Seadawy, Benjamin-Feir-instability in
nonlinear dispersive waves, Computers and Mathematics with
Applications 64 (2012) 3557-3568.
15- Seadawy, A.R., and El-Rashidy, K., Traveling wave solutions for some coupled nonlinear
evolution equations by using the direct algebraic method, Math.
and Comp. model. \textbf{57} (2013) 1371.
16- A. R. Seadawy, New exact solutions for the KdV equation with higher order nonlinearity
by using the variational method, Comp. and Math. Appl. 62 (2011)
3741-3755.
17- Seadawy A. R, Stability analysis solutions for nonlinear
three-dimensional modified Korteweg-de Vries-Zakharov-Kuznetsov
equation in a magnetized electron-positron plasma" Physica A:
Statistical Mechanics and its Applications Physica A 455 (2016)
44-51.
18- A.R. Seadawy, Nonlinear wave solutions of the three-dimensional
Zakharov–Kuznetsov–Burgers equation in dusty plasma, Physica A
439 (2015) 124–131.
19- A.R. Seadawy, Stability analysis of traveling wave
solutions for generalized coupled nonlinear KdV equations, Appl.
Math. Inf. Sci. 10 (1) (2016) 209–214.
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We know that when the number of nonlinear equations is the same as unknowns, we can simply find the solutions by inserting different starting points. I have 3 nonlinear equations with 4 unknowns, with some bound constraints. How can I see if there is a solution to the problem?
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Dear Peter
Yes, you are right.  I considered minimizing the distance as you said but because of bound constraints, there is no solution but if there is another function in stead of distance then there will be a solution within the bound constraints. I wonder if there is similar approach in nonlinear equations like SVD  in linear equations where you can assign different values for arbitrary parameter and find all possible solutions. It would be nice to find a solution within the bound constraints.
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Numerical solution to PDEs through different schemes (implicit or explicit) results in some error which are referred as dissipative or dispersive error for the solution of Hyperbolic or Parabolic PDE. How these are verified and minimised for more accurate results
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We solve governing partial differential equations numerically in order to get solutions to complex fluid mechanics problems. In classical CFD using finite differences we first convert the p.d.e. to its discrete version by replacing each partial derivative term by a difference term using forward, backward or central differencing in time and space. The off-node and off-time level variables (like ui+1 & un+1) can be expanded as Taylor Series. After algebraic manipulations we arrange on the LHS of equation the difference form of the p.d.e. and on the RHS the other terms due to Taylor series. These left out terms on RHS contain delta_t, delta_x, delta_y along with the partial derivative terms of varying powers. Normally the terms on RHS are called the ‘truncation error terms’ and are neglected and we solve for the LHS difference equation representing the original p.d.e. Obviously how accurate and what type of results we get depends on the magnitude and nature of the discarded truncation error terms on the RHS. For the sake of better (convenient) physical interpretation all the partial time-derivatives occurring on the RHS are transformed into equivalent spatial partial derivatives, so that the truncation terms now have delta_t, delta_x, delta_y, constants and only spatial partial derivatives of various orders. The complete formulation (RHS + LHS) is called the ‘MODIFIED EQUATION’. As stated above the accuracy and the nature of the solutions we get depend on the truncation terms on the RHS.
The ACCURACY of the solution depends on the order of the delta_t, delta_x, delta_y present in the first term on RHS having lowest order of partial derivative.  For example if we have delta_t and (delta_x)^2 terms in the very first term on RHS, our solution is said to be first order accurate in time and second order accurate in space.
The NATURE of the solution means whether the solution is DISSIPATIVE or DISPERSIVE. If the lowest order partial derivative term on the RHS is ‘even’ i.e. 2,4,6 etc. the solution is DISSIPATIVE in nature. Whereas if the lowest order partial derivative term on the RHS is ‘odd’ i.e. 1,3,5 etc. the solution is DISPERSIVE in nature. In a DISSIPATIVE solution the sharp gradients like shocks get smeared off due to DISSIPATION and appear as a ‘ramp’ rather than a ‘step’. If the solution on the other hand is DISPERSIVE, the solution gets violently wavy in the region of sharp gradients, for example at the corners of the shock solution. (See John D Anderson’s CFD Book page 236).
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We want to use HAM Method, a numerical method of solution to solve flow and heat transfer problems under certain boundary conditions.
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I have come across a work by Professor Pradeep Sidhdheswar of Bangalore Univesity along with his collaborators in China.  You may either mail him or refer to his work.  Best wishes, Shailendhra Karthikeyan.
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Static gravity, i.e. the Schwarzschild metric, depends mostly on the vacuum field equation: Ricci tensor = 0.  Using notation from https://en.wikipedia.org/wiki/Deriving_the_Schwarzschild_solution we get one thing from symmetry and the definition of metrics, a form in which there are two relevant coefficients, A(r) and B(r) (the time and space coefficients, roughly speaking).
From the Ricci tensor = 0 constraint, which has been given only vague physical interpretations, we get two things:
  1. A(r)B(r)=K where K is a constant.  From the limit at great distances where A->B->1, we see that in normal coordinates A(r)=1/B(r) (or -1/B(r) if you are including the sign in the coefficient).  This gives the relation between time dilation and radial length contraction (or spatial expansion).
  2. Substituting for B(r) we get a differential equation rA'=A(1-A) where ' means derivative and A=A(r).  This has a solution of the form (1-K/r)-1 where here K is a different constant, and the Newtonian limit gives the usual Schwarzschild time dilation factor 1/(1-2GM/rc2)1/2 and the inverse for radius.
As a friend showed a few months back, this leads to gravity which for a distant observer with a tether amounts to an inverse square force law, oddly enough.  But potential, obviously, is not Newtonian for any observer.
Suppose we want one of the two variations of potential to be the time dilation factor.  These are both approximately equal to the Schwarzschild factor for large r.  They are 1/(1-GM/rc2) and (1+GM/rc2).  Take the second one.  It implies, according to another question I posed https://www.researchgate.net/post/What_differential_equation_has_a_solution_of_the_form_Fx1_1_Kx-2 , that the differential equation in step 2 above must be something like rA'=2A(1-A).
That factor of 2 when backed through to the Ricci tensor means that it cannot be quite zero for small radii.  At that point I get lost trying to draw more definite conclusions.  For example, if we have a different equation:
Ricci tensor = X
Then what is X (probably a complicated function) such that we still have A=1/B, but we get rA'=2A(1-A).  It is a matter for someone who knows all the little summations and conventions by heart and is good at fudging.  Any takers?
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If your institute has a licence, I would recommend Mathematica (from Wolfram), and for the tensor manipulations, xAct.
Otherwise, I recommend Reduce, which is now free. Here you'll have to type the expressions for the Christoffel symbols, etc..
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How to use the finite volume method to solve a system of 7 DPEs (differential partial equations) of transport type?
The problem is written in the attached image (the G(C_2.0) function is non-linear).
Thank you very much in advance.
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thank you against for your response.
yes I know that th FV method is for the spacial discretization,  and I discretise the temrs on time bu finites difference method.
I think that I have a problem of discretization with the 2 first equations.
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Hello Everyone 
I have to solve a higher order coupled PDE with initial and boundary conditios. 
I have tried Matlab pdex4 and pde but could not as they dont allow higher derivatives wrt t. My equations look like as follows
d2v/dt2 = d4v/dx4 + F
d2w/dt2 =d2v/dx2 + F
In Matlab/pde apparently it doesnt allow higher derivative on left handside.
Can anyone please help me in this?
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Hello Muhammad Usman
I believe this problem can be solved analytically using the method of separation of variables. It can be shown that one set of the possible solutions are the infinite series:
V(x, t) = p(t) + Σ [a1*cos(2β2t) + a2*sin(2β2t)]*[b1*Cosh(βx)cos(βx)+ b2*Cosh(βx)sin(βx) + b3*Sinh(βx)cos(βx) + b4*Sinh(βx)sin(β x)]
w(x, t) = p(t) - (1/2)* Σ (1/ β2)* [a1*cos(2β2t) + a2*sin(2β2t)]*[ -b1*Sinh(βx)sin(βx)+ b2*Sinh(βx)cos(βx) - b3*Cosh(βx)sin(βx) + b4*Cosh(βx)cos(βx)]
Where d2p/dt2 = F(t) and the series runs from β = certain value to ∞ and the constants a1, a2, b1.....b4 are functions of. β
Four boundary conditions and need to applied to determine the constants b1, b2, b3, b4. When the boundary conditions are applied an eigen value equation will be obtained for determining the admissible values of β. The coefficients a1 and a2 can be determined as Fourier coefficients when the two initial conditions are applied. When some boundary values are non-zero the above solutions need to be suitably modified by adding linear function of x..Note that another solution set is also possible in which the time functions are replaced by exp(-2β2t)
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I am working on some model analysis, getting two diff equations and after I convert them into matrix form, I have equations looks like
[A][X]=C*(exp([B][X])-1)
where C is a constant and Both [A] and [B] both n*n matrices and are found out from two diff equations and boundary conditions.
Now my question is how am I able to find the value of [X], I am thinking using fixed point method, but this is in matrix form, could any one give me any clue or hint?
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Try quasilinear methods as published by V. Mandelzweig et al., which can be adapted to nonlinear matrix equations.
The basic idea therein is based on Newton-Cotes method.
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Here is a nonlinear equation for beam on elastic foundation
d''''(y)/dx''''-D*d''y/dx''+k*y=f(x)
I've solved above equation with the aid of Laplace Transform. The closed form equations are obtained.
I'm going to solve the following equation which has a nonlinear term:
d''''(y)/dx''''-D*d''y/dx''+k*[y/(1+G*y)]=f(x)
f(x)=FF for x<L
f(x)=0 for x>L
How could I solve this nonlinear equation?
comment:
1- d'x stands for 1st derivative of f with respect to x
2- D, k and G are constants
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Depending on your application you could also use Timoshenko beam equation which is just a second order ode.
In principle I would use a numerical solution, either to solve Euler-Bernoulli, or Timoshenko beam eqs, i.e. by employing a finite differences scheme.
I could suggest some references if you want to go with the numerical approach, not sure if this is what you want though. 
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I would like to know how two partial differential equations such as advection-dispersion-reaction equation and first-order kinetic mass transfer equation in contaminant transport modeling are solved together numerically. The example that I have been involved with is in the manual of MT3DMS code (Zheng and Wang, 1999), page 52. The two equations are combined into the coefficients of one matrix equation, but I cannot understand if the two partial differential equations are first merged together and then the coefficients of the matrix are obtained, why none of the parameters of the model are not eliminated, in oppose to what we expect from the merged equations?
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