Partial Differential Equations

Partial Differential Equations

  • Dmitry Gromov added an answer:
    Is it possible to apply dynamic programming to optimal control problems with fixed right end-point?

    In this case, instead of the boundary condition V(Tf) =0 we have V(Tf,xf)=0 which is not enough to solve the related PDE.

    I wonder if it is possible to extend/modify the classical formulation to come up with a well posed problem. Obviously, we assume that the system is controllable and it is always possible to drive the system to the desired end-point xt.

    Dmitry Gromov · Saint Petersburg State University

    In dynamical programming, V(t,x) is the optimal value of the cost function that we get when starting at time t from point x. And, respectively, V(tf,xf)=0 is not a termination condition, but the boundary condition for the Hamilton-Jacobi-Bellman partial-differential equation which I have to solve in order to get V. This condition says that when starting at final time=tf from the final point=xf, the best reward we can get is equal to 0.

    Thus, the described approach seems to be not very relevant to the problem described.

  • Waldemar W Koczkodaj added an answer:
    How can I construct the system of equations?

    How to construct the system of equations by susbtituting the assumed series solution in the nonlinear partial differential equation? Kindly find the attachment.  

    Waldemar W Koczkodaj · Laurentian University

    Sorry. It is too complicated for me but I am glad that you have been helped.

  • Chithra K. P. added an answer:
    How can I take the convolution Sumudu transform for the same function?

    Example : sumudu transform of f^2(t).

    f(t) is unknown

    Chithra K. P. · Not Affiliated

    Please see the documents given in the following links.

    http://www.emis.de/journals/HOA/JAMSA/2006/91083.pdf

    http://www.jpier.org/PIER/pier74/17.07050904.Hussain.B.pdf

    http://www.ece.ualberta.ca/~sumudu/SumuduTransform.pdf

  • Sanjiv Sharma added an answer:
    What is an inverse scattering transform and how can it be used for source separation techniques?

    Please guide me to a source which builds the idea from the basics.

    Sanjiv Sharma · Airbus, UK, Filton

    You may also consider the paper by Ablowitz and Clarkson:

    Ablowitz, M. J. and P. A. Clarkson (1991), Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society lecture Notes 149, Cambridge University press.

    When you have developed a richer understanding, may I encourage you share it by contributing to http://www.scholarpedia.org/article/Inverse_scattering_transform_method?

  • Elkhan N. Sabziev added an answer:
    Has anyone used the fractional differential transformation method (fDTM) in order to solve the linear partial differential equations?

    Has anyone used the fractional differential transformation method (fDTM) in order to solve the linear partial differential equations?

    Elkhan N. Sabziev · Kiber Ltd Company

    Unfortunately, I longer not do this topic ......
    I believe that in your University or at the nearby University you can find mathematician from Azerbaijan. I hope they will help you understand 4 main pages. Please, download next pdf-file and see page 610-613: 
    "http://inis.jinr.ru/sl/vol2/Mathematics/%D0%90%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7/%D0%A1%D0%B0%D0%BC%D0%BA%D0%BE,%D0%9A%D0%B8%D0%BB%D0%B1%D0%B0%D1%81,%D0%9C%D0%B0%D1%80%D0%B8%D1%87%D0%B5%D0%B2,_%D0%98%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB%D1%8B_%D0%B8_%D0%BF%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%BE%D0%B4%D0%BD%D1%8B%D0%B5_%D0%B4%D1%80%D0%BE%D0%B1%D0%BD%D0%BE%D0%B3%D0%BE_%D0%BF%D0%BE%D1%80%D1%8F%D0%B4%D0%BA%D0%B0,1987.pdf"

  • Majeed Ahmad Yousif added an answer:
    Is the Homotopy Analysis method suitable for solving a hyperbolic system of first order partial differential equations?
    I am modeling flow of fluid through elastic tubes.
    Majeed Ahmad Yousif · University of Zakho

    You can but Homotopy perturbation method is more suitable suitable to solving a hyperbolic system of first order partial differential equations

    see this article papers

  • Muhammad Mujtaba Shaikh added an answer:
    Why are analytical solutions of the Navier-Stokes equations not valid for most fluid flow situations?

    Why are analytical solutions of the Navier-Stokes equations not valid for most fluid flow situations?

    Muhammad Mujtaba Shaikh · Mehran University of Engineering and Technology

    As not all fluid flow situations are as simple and linear to be treated with analytically, so NS-equations don't have general analytic solution. However under some specifications, one can think to solve these analytically. Nevertheless the process is very complex.

  • Thomas I. Seidman added an answer:
    When does a first order partial differential equation in the form dh(x)/dx*g(x)=1, where h(x) is scalar and x belongs to R^n, admit solution?

    As said in the title: under which hypotheses the solution of dh(x)/dx*g(x)=1 exists?Where h(x) is a scalar function, g(x) is a function from R^n into R^n and x belongs to R^n.

    Thomas I. Seidman · University of Maryland, Baltimore County

    Let H=grad h (i.e., del h).  Then I would interpret the condition as asking that the dot product of H with g should everywhere be 1.  At each x this asks that the vector H(x) lies in a certain affine subspace, a geometric condition.  Of course, the other important condition is that the vector field H must be a gradient field.  For the method of characteristics to apply one would consider a trajectory x(t) -- starting at a point where the given equation holds -- and then look to see if the condition on H.g could hold for all (?) variations dx.  This doesn't solve the problem, but may give some clarification/direction. 

  • Ali Azimzadeh added an answer:
    Why does the error message "Too many attempts made for this increment " occur when XFEM analysis is done in Abaqus for a cracked tension specimen ?

    Model Properties

    Aluminium, 2D plane stress model with an edge crack of 8mm in length at one edge.

    Ali Azimzadeh · Azarbaijan Shahid Madani University

    Dear Harilal 

    if you have loading depends on time, check your step module.in "basic" section and "Automatic stabilization" subsection, if you've chosen "None" or other thing, edit it and  choose "specify damping factor". then analyze your model again. maybe this is your modeling problem.

  • Toufic El Arwadi added an answer:
    How can I solve the Modified Bessel PDE using FEM?

    I am attempting to solve a modified bessel partial differential equation using a finite element implementation. I have used FDM (with reasonable success) and was hoping to replicate the results using FEM. My exposure to FEM is limited and I was hoping a commercial suite (such as Matlab or Mathematica) could solve this problem. My problem is defined as such - 

    u(r,t)=0 at t=0 and r=0

    du/dr=k1*u(r,t) at r==1  where k1 is a constant

    d2u/dr2 +1/r*du/dr - u/r^2 = du/dt +r/2*k2, where k2 is a constant

    u and u(r,t) are interchange-able in notation

    My question is - is this problem solvable using an FEM toolbox?

    Many thanks. RM

    Toufic El Arwadi · Beirut Arab University

    Yes it is possible, you have different ways : first approximate du/dt by using the classical approximation (u^{n+1}(r)-u^n(r))/dt than write the semidiscrete scheme (most famous explicit, implicit scheme). After this write the variational problem and go to FreeFem++ to implement it 

  • Lukas Mosser added an answer:
    Can anyone help me to solve a system of partial differential equations iteratively or by mixed-fem formulation?

    I would like to solve the following system (Equation 17 in the link) of linear partial differential equations using the finite element method. In two dimensions we have a total of 3 unknowns per node. Two for the displacements and 1 for the scalar fluid pressure.

    Is it possible to do this in an iterative way using the standard FEM approach?

    Simple Schematic

    1. Calculate displacements from initial and boundary conditions from the first equation K*u - Q*p = f_u 
    2. Calculate new pressure from the second equation by plugging in the displacements from the previous calculation.

    3. Repeat 1 and 2 until there is only small changes in the unknowns.

    4. Go to the next time step and repeat procedure from 1.

    Or will I have to use a mixed finite element formulation where I solve for displacements and pressure simultaneously?

    If you could lead me into a direction or provide some help it would be greatly appreciated. Thank you!

    Lukas Mosser · Montanuniversität Leoben

    Dear all,

    Thank you very much for your comments they have been very helpful and have led me into a number of interesting and possibly fruitful directions.

    Since this is an ongoing project, I will let you know what I used as a solution when I have managed to implement everything.

    Best regards,

    Lukas

  • How can we use COMSOL to solve ''third order and up'' of partial differential equations?

    solving third order and up of PDEs.using Comsol

    Waranatha L Abeygunasekara · University of Peradeniya

    With comsol, if you need to solve 'third order and up' differential equations, you have to convert them into set of second order  and first order differential equations first. I  guess the process of converting a nth order equation into n number of first order equations is a standard procedure. you can check this out at any standard mathematics text.

  • Mehmet Yavuz added an answer:
    How can I obtain the fractional Black-Scholes and the generalized fractional B-S Equation?

    By considering the Black-Scholes equation, how can I obtain the fractional Black-Scholes Equation and generalized fractional Black-Scholes Equation?

    Mehmet Yavuz · Necmettin Erbakan Üniversitesi

    Dear Khaliq, a study related the question in the attachment...

  • Yehia A. Khulief added an answer:
    Higher modes truncation effect of a PDE's solution?

    When solving a PDE using the method of separation of variables, some higher modes are eventually truncated and not considered.
    How significant is the effect of higher modes that are not considered?
    Some believe adding a constant feed-through term will solve the problem, is there any other more accurate method to compensate this negative effect of truncation?
    *. Considering the fact that this is an engineering problem not a pure theoretical math problem.

    Yehia A. Khulief · King Fahd University of Petroleum and Minerals

    Let me address this controversial question within the context of vibration analysis of continuous systems (hyperbolic PDE). When we represent the system dynamics in the modal space, we use a finite (truncated) set of modes (commonly called significant modes), which include the low-frequency subsystem. The accuracy of the solution is mainly dependent on the percentage of the system's kinetic energy contained in this selected set of modes. The difficult question has always been the following: What is the appropriate number of significant modes? In fact, there is no a priori means to judge the dimension of the significant frequency subsystem. Yet, there are some indirect measures to give some insight into this problem. (1) if the excitation is smooth, then very likely most of the system kinetic energy will be picked up by the first few modes. the term "few" may widely vary for different systems. A couple of simulation runs would lead to the appropriate number of modes as judged by convergence. (2) the tricky case is the related to the impulsive excitation; i.e. due to impact for instance. In this case, higher modes are normally excited to pick up proportionally and appreciable percentage of the system's kinetic energy. In this case, more higher modes need to be included in the model. Since there is no readily available information as to which  higher modes will be excited, one must perform few preliminary runs to check convergence as the size of the frequency spectrum is increased. It is also possible to find out which higher frequencies may be excited by the impulsive force if the frequency spectrum of the excitation is available. 

  • Alexander Pchelintsev added an answer:
    How many types of methods are there to convert partial differential equation into an ordinary differential equation?

    By the differential equation we can solve any problem of any section in mechanics. So, for that I want to know.

    Alexander Pchelintsev · Tambov State Technical University

    You present one of the partial derivatives on the formula of finite differences. You will have a system of ordinary differential equations. A count of equations in the system will depend of step.

  • Majeed Ahmad Yousif added an answer:
    Is there any numerical solution to solve nonlinear coupled PDEs?

    Is there any numerical solution to solve nonlinear coupled PDEs?

    Majeed Ahmad Yousif · University of Zakho

    use homotopy perturbation method

  • Mohamed El Naschie added an answer:
    Why when we use Homotopy Perturbation Method for solving nonlinear PDEs, at increasing time, approximate solution away to exact solution?

    when increase time will increase absolute error.

    Mohamed El Naschie · Alexandria University Alexanderia Egypt

    ask prof je Huan-HE.

  • Mohamed El Naschie added an answer:
    What is the relation of representations of su (2) and generating functions of partial differential equations?

    Lie algebra, Casimir element, special functions, partial differential equations

    Mohamed El Naschie · Alexandria University Alexanderia Egypt

     lw plus w minus and z zero of the weak force is all i need of us2 in my work in the stander modell

  • Meguenni Bouhadjar added an answer:
    Can anybody suggest me the best software for Partial Differential Equations (PDEs) ?

    I want to solve partial differential equations (PDEs), which contains both space (x) and time (t). What is the best software for this purpose?

    I also want to know the most appropriate numerical algorithm for this so that I can write a program to solve PDEs.

    All types of suggestions are highly appreciable.

    Thank You.

    Meguenni Bouhadjar · Agence Spatiale Algérienne

    I think Octave is the best (its Free), but if you are interested to find analytical solutions, you can use Maple.

    For Octave see: http://waxworksmath.com/Software/pdeone_rkc.html

  • Mostafa Saeedi added an answer:
    How can one solve a set of PDE and ODE simultaneously in MATLAB ?

    I hope you are fine

    I have some problems with my codes in MATLAB software, and I wish you help me, I should solve set of PDE and ODE's in MATLAB which are related to each other. The set of PDE includes 4 elliptic PDE with coefficients is related to some ODE and I can't solve them If you could help me please guide me as well.

    Thanks a lot and waiting for your prompt reply by return.

    Regards M.Sheikhi

    Mostafa Saeedi · Iran University of Science and Technology

    محمد در کنار فلودایزبد عکس گرفته..........

    الکی مثلا محمد مهندس طراح شده

  • Sofiane El-Hadi Miri added an answer:
    Is there a comparison principle for anisotropic operators?

    I'll be very grateful if you can tell me if there exists a comparison principle for the following anisotropic operator :

     -∑∂_{i}[|∂_{i}u|^{p_{i}-2}∂_{i}u]

    Thank you for your help.

    Sofiane El-Hadi Miri · Abou Bakr Belkaid University of Tlemcen

    Thank you for your useful answers .... deep regards.

  • Alejandro Sarria added an answer:
    Is there any relationship between curvature and eifenforms of D'Alembert operator?

    Global analysis

  • David Mark Koenig added an answer:
    What is the difference between symmetric and asymmetric eigenvalues?

    On pp. 206-212 in Nowacki's "Dynamics of Elastics Systems", a pde is solved for the transverse vibrations of a rectangular plate subject to clamped boundaries. Two equations are developed for the eigenvalues - one for the "symmetric" modes of vibration and one for the "asymmetric" modes. Although I can follow Nowacki's development, I do not understand his use of "symmetric" and "asymmetric". Can someone explain the use of these terms? Thank you.

    David Mark Koenig · Corning Incorporated

    Thank you.

  • Rogier Brussee added an answer:
    What is the meaning of the Laplace operator whose one eigenvalue is 1?

    This is related to global analysis.

    Rogier Brussee · Hogeschool van Utrecht

    Although apparently a 1 makes sense in the discrete context,  I don't really think it makes any sense on differential manifolds. First lets assume that the manifold is compact, as otherwise the spectrum does not consist of eigenvalues. Moreover, the spectrum is unbounded so that there is not a unique eigenvalue. Finally in thecompact case there is a smallest non zero eigenvalue, but one can change the eigenvalue by scaling the metric. The proper formulation therefore must involve a  normalisation, like volume = 1, or better, something that scales with the metric.  It seems to be closely  related to the isoperimetric inequality as one has

                   h^2(M)/4  <= \lambda_1 <= C(dim(M)) (h^2(M) + h(M)\delta )

    where \delta is a lower bound on the Ricci curvature and 

                 h(M) = Cheeger constant
                         = inf_{D\subset M, vol(D)\le vol(M)/2 } ( vol (D)/vol(\boundary D) )

    see A note on the first nonzero eigenvalue of the laplacian acting on P-forms
    Bruno Colbois, Gilles Courtois, see manuscripta mathematica
    1990, Volume 68, Issue 1, pp 143-160, and references therein.
                                                                 

  • Daniil Yurchenko added an answer:
    Is it possible to find a super-solution to the Hamilton-Jacobi-Bellman equation?

    In the paper of Ishii and Koike "Boundary regularity and uniqueness for and elliptic equation with gradient constraint", they guarantee a solution to the following equation  

    max{Lu-f, |du|-g}=0 a.e. in Omega (this set is a bounded domain in Rn),           (1)

                             u=0 on the boundary of Omega (the boundary has some                                                                                              regularity),

    where L is an elliptic operator of the form Lu=-aijuxi xj+bjuxj+cu, c is bigger or equal than zero and |du| denotes a size of the gradient of u of the form |du|2=dijuxjuxi. Assuming that aij, bj, dij, g in C2(closed of Omega), c,f in C1,alpha(closure of Omega), aij in C3,alpha(boundary of Omega) and there exists a function w in C1(closure Omega) intersection with W2.ploc(Omega) with p>n satisfying

    max{Lw-f, |dw|-g} is less or equal than 0 a.e. in Omega,                                    (2)

    u=0 on the boundary of Omega,

    then,  they obtain that there exists a solution u to the HJB equation (1). If g>0 and c>0 or f is bigger or equal than zero, the solution is unique in C(closure Omega) intersection with W2,ploc(Omega). My question    

    My question is: it is possible to find a function w that satisfies (2), in the case when we only know that f is in C1,alpha(closure Omega)?   

    Daniil Yurchenko · Heriot-Watt University

    If you want to find actually an explicit expression for the function that satisfies your HJB equation, then you can try the approach I used (for a degenerate parabolic type HJB, related to a stochastic vibrational system) which is based on the method of characteristics or the quadratic form, depending on the type of equation you have. If you want to find some estimates or to prove the uniq or exist then I am not sure you have to ask mathematicians.

    Good luck.

  • Ulrich Mutze added an answer:
    How can solve Simulteneous Partial Differential equations?

    How can solve Simulteneous Partial Differential equations?

    Dr. Sunil Kumar

    Sunil,

    as most of the previous answers made clear it is very unlikely to find exact solutions to an arbitrary system of partial differential equations. Neverthess there are cases where the solutions are known and play a major role in science (e.g. cases of Maxwell's equations). Actually, there exists a huge body of knowledge on special systems for which exact solutions are known. When I was a student the books by Erich Kamke were the established sources for finding out whether for a given equations there is something known about exact solutions. Today more convenient and probably more complete information is collected in the computer algebra systems Mathematica and Maple.

    A completely different issue are numerical solutions. Here there are methods known that work even on mixed differential/algebraic equations. Here the chance to get useful results from the above-mentioned computer algebra systems is really large.

  • Shima Doorandish added an answer:
    How can I solve "unknown function or operator" error in comsol ?

    Hello, I used Mathematics>PDE interfaces in comsol. I specified parameters and then I wrote degradation formula in variables part and so on.

    At the end, when I pressed "compute", I got this error :

    -Unknown function or operator
    -Name:b
    -Feature:Time-Dependent Solver1 (sol1/t1)
    -Error:Unknown Function or operator

    I attached the file here.
    Thank you so much for your help

    Shima Doorandish · University of Tehran

    Thank you so much dear Hossein , I was really confused.
    Yes , it works .

    Thanks a lot.

  • Belkacem Kada added an answer:
    Can anybody recommend a literature or review of infinite dimensional sliding modes?

    Hi all, can somebody help me by pointing out any literature survey or a review article in the field of infinite dimensional sliding mode control (sliding mode control for infinite dimensional systems, those described by PDEs)?

    Belkacem Kada · King Abdulaziz University

    Dear Imran Khan Yousufzai

    You can download my paper " Family of Dynamic Solutions for Real-time Control of Distributed Parameter Systems" from www.researchgate.net . It would help you in understanding some concepts about the control of DPSs

  • Phu Anh Phi Nghiem added an answer:
    How to calculate the analytic solution to 3D acoustic wave equation in a layered media?

    Wave equation

    Wave propagation

    Numerical methods

    Wave simulation

    Phu Anh Phi Nghiem · Atomic Energy and Alternative Energies Commission

    Dear Wensheng,

    I developped a simple semi-analytic method for calculating acoustic modes in a layered media but with spherical symmetry, applicable to stars for example (helio- or astero-seismology). Attached is this article. I don't know if this can help?

    Best regards,

    Phi

  • Jibin Raj added an answer:
    How are the boundary conditions incorporated while using the Differential Quadrature method?

    When dealing with 2-D Laplace equation how is Differential quadrature method applied to get the solution. How are the four boundary conditions incorporated into the formulation?

    Jibin Raj · SRM University

    Sir I got the solution for the 2-D Poisson equation and is matching with the analytic solution.

    Now I am trying to solve the free vibration of a cantilever beam. I am attaching a word document in which I have given the progress of my work. I am not able to converge to fundamental frequency. I have tried increasing the sample points but still I am not getting the result. Can you please refer my work and advise me where I have gone wrong.

    I have referred the review paper of C.W.Bert and Moinuddin Malik (page No 10).

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