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# Nonlinear Partial Differential Equations - Science topic

Explore the latest questions and answers in Nonlinear Partial Differential Equations, and find Nonlinear Partial Differential Equations experts.

Questions related to Nonlinear Partial Differential Equations

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.

**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.*

what are the mostly used and applicable existence and uniqueness theorem for nonlinear partial order differential equations?

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 ?

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?".

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?

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.

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.

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

- 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.

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.

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.

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.

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

**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!**

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

**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!**

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

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?

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.

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.

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!

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!

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?

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.

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

- 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.

- 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.

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.

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?

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.

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?

Is BVPh Mathematica package valid for linear operator with respect to "t" derivative in one dimensional PDEs by Homotopy analysis method?

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

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

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.

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?

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.

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.

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.

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

I am trying to derive weak-field Schwarzschild metric using Linearized Einstein's field equations of gravity:

[]h

_{μν}– 1/2 η_{μν}[]h = -16πG/ c4 TμνFor static, spherically symmetrical case, the Energy- momentum tensor:

T

_{μν}= diag { ρc^{2}, 0, 0, 0 }Corresponding metric perturbations for static ortho-normal coordinates:

h

_{μν}= diag { h_{tt}, h_{xx}, h_{yy}, h_{zz}}With one index rised using flat space-time Minquoskwi metric η

^{μν}= { -1 , 1, 1, 1 }:h

^{μ}_{ν}= diag { -h_{tt}, h_{xx}, h_{yy}, h_{zz}}Trace of the metric:

h = h

^{γ}_{γ}= - h_{tt}+ h_{xx}+ h_{yy}+ h_{zz}The four equations:

1) []h

_{tt}– 1/2 η^{tt}[]h = -16πG/ c^{4}T_{tt}=> []h

_{tt}+ 1/2 []( - h_{tt}+ h_{xx}+ h_{yy}+ h_{zz})= -16πGρ/ c^{2}=> 1/2 []( h

_{tt}+ h_{xx}+ h_{yy}+ h_{zz})= -16πGρ/ c^{2}2) []h

_{xx}– 1/2 η^{xx}[]h = -16πG/ c^{4}T_{xx}=> []h

_{xx}- 1/2 []( - h_{tt}+ h_{xx}+ h_{yy}+ h_{zz})= 0=> 1/2 []( h

_{tt}+ h_{xx}- h_{yy}- h_{zz})= 0Similarly:

3) 1/2 []( h

_{tt}- h_{xx}+ h_{yy}- h_{zz})= 04) 1/2 []( h

_{tt}- h_{xx}- h_{yy}+ h_{zz})= 0Adding equations 2), 3) & 4) to 1) respectively, yield:

[]( h

_{tt}+ h_{xx}) = []( h_{tt}+ h_{yy}) = []( h_{tt}+ h_{zz})= -16πGρ/ c^{2}Solving the equations using:

[] ≈ ▼

^{2}≈ 1/R^{2}d/ dR ( R^{2}d/ dR ) for static spherically symmetric case; we get:( h

_{tt}+ h_{xx}) = ( h_{tt}+ h_{yy}) = ( h_{tt}+ h_{zz})= -8πGρR^{2}/ 3c^{2}– K_{1}/ R + K_{2}Similar solutions for vacuum case, with Tμν= 0 would be:

( h

_{tt}+ h_{xx}) = ( h_{tt}+ h_{yy}) = ( h_{tt}+ h_{zz})= – K'_{1}/ R + K'_{2}For the metric to be asymptotically flat:

K

_{2}= K'_{2}= 0For the solutions to be continuous at boundary, R= r, the radius of spherically symmetric matter:

- 8πGρr

^{2}/ 3c^{2}≈ - 2Gm/ rc^{2}The remaining two constants must be:

K

_{1}= 0 & K'_{1}= 2Gm/ Rc^{2}Therefore, my solution comes:

( h

_{tt}+ h_{xx}) = ( h_{tt}+ h_{yy}) = ( h_{tt}+ h_{zz})= - 2Gm/ Rc^{2}But, as per the literature, the weak field Schwarzschild metric must come out to be:

h

_{tt}= h_{xx}= h_{yy}= h_{zz}= 2Gm/ Rc^{2}Thus the solutions must come out to be:

( h

_{tt}+ h_{xx}) = ( h_{tt}+ h_{yy}) = ( h_{tt}+ h_{zz})= + 4Gm/ Rc^{2}I am not able to make out where I am making mistake. Can anybody please help?

Thanks.

Nikhil

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.

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 .

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

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!)

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.

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

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 ;)

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!

E. H Doha

M. A. Abdelkawy

A. H. Bhrawy, or any other who knows the answer

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!

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.

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.

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(n

_{x}x n_{x}),B(n_{x}x n_{u}),C(n_{y}x n_{x}) and T(n_{x}x (n_{x}+n_{p})) - variables matrices, Q(n_{y}x n_{p}), F(n_{u}x (n_{x}+n_{p})), S(n_{x}+x_{p}x n_{x}+n_{p}) - 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 n

_{x}<n_{p}.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?

We are interested in computing the largest Lyapunov exponent for a variant of the the Kurimoto-Sivashinsky equation in 1+1 dimensions. Any suggestions?

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?

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.

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

My function values are going negative while i need to restrict it to zero. How do i do that?(optimal control problem)

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?

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

We want to use HAM Method, a numerical method of solution to solve flow and heat transfer problems under certain boundary conditions.

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:

- 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).
- 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/rc^{2})^{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/rc

^{2}) and (1+GM/rc^{2}). 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?

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.

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

d

^{2}v/dt^{2}= d^{4}v/dx^{4}+ Fd

^{2}w/dt^{2}=d^{2}v/dx^{2}+ FIn Matlab/pde apparently it doesnt allow higher derivative on left handside.

Can anyone please help me in this?

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?

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

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?