Partial Differential Equations

Partial Differential Equations

  • 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

    dear friend-  I am serious that you should ask prof je= Huan He..... he will answer you....he is a nice modest man

  • V. G. Marikhin added an answer:
    Is anyone familiar with transformations of diagonal Laplacians?

    May be who knows about the tranformations from one diagonal Laplacian to another diagonal Laplacian (change of variables)? Especially I'm interested in a case of transformation of Cartesian Laplacian to diagonal Laplacian with rational coefficients in 3D case.

    V. G. Marikhin · Russian Academy of Sciences

    Thank you, Magda. I hope your answer helps me.

  • Thomas I. Seidman added an answer:
    Hi everybody. Does anyone know if i can solve numerically the 2-D pure advection equation with the Galerkin Method?

    I am working on my Thesis (Bachelorarbeit), and I am having trouble solving a simple equation, because of the non symmetric advective matrix.

    My thesis is about a 2-D pollution transport system without diffusion.

    The transport equation is:
    dc/dt=-v*∇C


    After applying Galerkin Method, using nodes in the vertex of a rectangle.


    The weak form, at the end is

    [M]*C°+[K]*C=0,

    where C° = dc/dt.
    K = Advective Matrix.
    M = Mass Matrix.
    C= Concentration


    For example, for one Dimension , [K ]is

    K = U/2* [-1 1
    -1 1]

    After global assembling of my system, the result are zeros in the diagonal, and there is nothing that i can do. to find the Concentration in the unknown nodes.

    I have read a lot about stabilization methods, like Petrov Galerkin or Artificial Diffusion, but all require a little of Diffusion, specifically to determine the Peclet Number (Pe), but i have none Diffusion.

    I hope someone can help me, how to proceed.

    Greetings from Chile
    Juan Ignacio Correa.

    Thomas I. Seidman · University of Maryland, Baltimore County

    If I correctly understood your question (I am somewhat confused by the distinction between c and C), you are looking at a linear 1-st order system dc/dt=-v*c so the relevant theory is the Method of Characteristics.  This gives a family of linear 1-st order ODEs along the characteristics (straight lines if v is a constant vector). You have said nothing about boundary/initial data, but you must specify some data for each of these lines, typically the value at the intersection of that line with an initial surface.  Again assuming I have correctly understood the problem, this should be easy to solve explicitly.

  • Alzaki Fadlallah added an answer:
    What is the name of this inequality?

    (a+b)^{p}<=2^{p-1}(a^{p}+b^{p}) where P>=1 and a,b>0

    Alzaki Fadlallah · University of Alabama at Birmingham

    Thank you 

  • A. J. Roberts added an answer:
    Please. help! Is there any method to solve this nonlinear PDE using similarity?

    I stumbled upon equation j and l in my research.

    I found some books related to fluid dynamics, it says solution to j is the formula k

    (it didn't give the procedure, just solution)

    as you can see j and l looks similar. and similar boundary condition and mass balance.

    Assuming that solution to j is k, Is there any method to solve l and get the formula of h2?

    I tried to solve j analytically by using separation of variables, but it gives second order nonlinear ODE; and I made Wolfram Alpha solve it. but still it didn't respond. 

    so I decided to circumvent reaching the solution by using the similarity

    A. J. Roberts · University of Adelaide

    Regarding the comment "It is likely that some of the assumptions use to derive its equations break down for very small t, where this solution becomes singular." True.  But for a range of similarity problems one can prove that the similarity solution emerges from quite general initial conditions---including ones that are within the domain of validity of the derivation of the equations.  That is, often the similarity solution is widely relevant irrespective of its notional singularity at time zero. 

    For one example of this see the paper   S. A. Suslov and A. J. Roberts. Similarity, attraction and initial conditions in an example of nonlinear diffusion. J. Austral. Math. Soc. B, 40(E):E1–E26, Oct. 1998. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/453.

  • Emanuel Guariglia added an answer:
    Is Fractional Calculus useful in every branch of science?
    I.e. Applications of fractional calculus.
    Emanuel Guariglia · University of Salerno, Italy

    Of course. It has many applications in different fields of the science: nowadays its applications in economy, signal processing and image processing are among the most interesting. Fractional calculus can be applied also to other fields of Mathematics: for instance, I used it to investigate in the theory of special function. If you are interested, you can read my attached article on it.

  • Shaibu Mohammed added an answer:
    What methods exist about finding exact solution of nonlinear partial differential equations?

    Any suggestion/resources are appreciated.

    Thank you so much.

    Shaibu Mohammed · University of Aberdeen

    Ali M. Hourina: I am very grateful for this book. I have for long be searching for a comprenhesible book on nonlinear partialdifferential equations. Thank you one again.

  • Xiao'long Zhang added an answer:
    CAn anyone help with HAM Method?

    In HAM method first i divide the linear part and non-linear parts after that i get confused how can proceed for further solution process please explain me any body.

    Xiao'long Zhang · Dalian University of Technology

    This paper may be useful 

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

    This is related to global analysis.

    Cenap Özel · Dokuz Eylul University

    who will send you, profesor sergeyyy :)

  • Junior Barrett added an answer:
    How do I solve higher order coupled PDE's?

    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?

    Junior Barrett · Bacon's College England

    Could convert to one equation  with variable coefficients, but may prove difficult from there. Power series may work as a result if the series converges.

  • Amaechi J. Anyaegbunam added an answer:
    Do you have solved examples of system of nonlinear pde using finite element method?

    See above

    Amaechi J. Anyaegbunam · University of Nigeria

    Hello Milind M. Mathapati

      In order to understand how to solve a system of coupled nonlinear partial differential equations refer to pages 37-39 of the text book," I. M. Smith and D.V. Griffiths (1988). Programming the Finite Element Method, 2nd ed., John Wiley & Sons "

    Therein is explained how the steady state coupled 2D nonlinear Navier-Stokes equations is discretized using the continuous Galerkin FEM with quasi-linearization of the nonlinear terms. The Navier-Stokes PDEs couples u,v and P. You can discretize your own coupled system via analogy.

    I hope you will find this suggestion helpful.

  • Vahid Mosallanejad added an answer:
    Is there any numerical solution to solve nonlinear coupled PDEs?

    Is there any numerical solution to solve nonlinear coupled PDEs?

    Vahid Mosallanejad · University of Science and Technology of China

    Hi, Mrs. Mohsen

    Yes, there are quite different methods. I believe COMSOL software make big progresses for solving system of PDE's. you can use module "Mathematic", "Coefficient Form of PDEs". there is a tutorial video about that in internet.  As Mrs. samuli  described, there are other software as well. I like to add MATLAB too. If you want to learn about numerical method I recommend you to focus first on Newton-Raphson method which is involve making Jacobian matrix, because it is very fast. If you are going to write a code( instead od using software) it is better to first select appropriate solution method which is depend on the type of PDE( like, Eigen value PDE or Time dependency, or number of dimension) then try to implement different steps of an individual method.

    good luck

  • Majeed Ahmad Yousif added an answer:
    How can I solve a nonlinear partial differential equation [U_t=Q(t)*(1+(U_x)^2)^(-1/2))] using MATLAB?

    Can you suggest how to (analytically or numerically) solve this PDE?

    Majeed Ahmad Yousif · University of Zakho

    You can use Mathematica program

  • Majeed Ahmad Yousif added an answer:
    What are the best perturbation methods for nonlinear PDEs ?
    An example would be the nonlinear heat diffusion equation
    Majeed Ahmad Yousif · University of Zakho

    You can use use the best method SHAM

    you can get attached files, all of the are not  open acces

  • Thomas I. Seidman 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.

    Thomas I. Seidman · University of Maryland, Baltimore County

    I note that the video suggested by Florian Munteanu describes (a special case of) similarity solutions.  These are related to deep properties of the PDEs (and of the underlying physics) in considering structure preserving transformations: symmetries leaving the PDE invariant).  I note as particularly important the search for traveling wave solutions leading to d'Alembert's solution of the wave equation and, for a nonlinear example, to solitons.

  • Elemer Elad Rosinger added an answer:
    How can we determine a better function space to solve PDEs?

    I work on function spaces, and i want to see applications and how can we take them to find a better solution for our equation. 

    Elemer Elad Rosinger · University of Pretoria

    Dear Christoph : You are of course right. However, that is NOT the point. NO, the point is that we ARE far too near associated with monkeys ... 
    And to make it more clear to you, here is a classic related joke :
    The little boy comes home from school and enthusiastically tells his father :
    Father ! Do you know what we learned today in school ?
    What ?, ask the father.
    To which the boy replies that : We come from monkeys !
    To which his father replies : You DO, but I DON"T !!!

    Gotcha ???

  • Mehdi Amiri added an answer:
    How do you solve coupled 1D time independent partial differential equation using comsol or matlab?

    here i am attaching comsol model which i desgined to solve these two coupled partial differential equation,but not getting correct results 

    du1/dx=-1.148*u1-4.462*u1*u2-3.8*10^-12*u1^2-1.9*10^-22*u1^3-6.83*10^-22*u2*u1^2-2.36*10^-22*u1*u2^2.........(1)

    du2/dx=-1.148*u2+50*u1*u2-4.93*10^-12*u2^2-3.41*10^-23*u2^3-2.44*10^-23*u2*u1^2-11.17*10^-23*u1*u2^2........(2)

    initial conditions

    u1(0)=1,u2(0)=.01 

    Mehdi Amiri · University of Maryland, College Park

    That would be great Amit, I appreciate your help!

    Mehdi

  • Di Yang added an answer:
    Are there evolution equations admitting peakon solutions?

    It is well known that certain integrable equations such as the Camassa-Holm, Novikov and Qiao admit the so-called peakon solutions. However, all of them are not evolution equations. Then I would like to know if there is some evolution equation admitting such property.

    Di Yang · Scuola Internazionale Superiore di Studi Avanzati di Trieste

    The answer can be positive. Indeed, Camassa-Holm equation can be considered as a (local) evolution equation by taking inverse of the operator 1-\partial_x^2. Note that (1-\partial_x^2)^{-1}=1+\partial_x^2+\partial_x^4+\dots. But, of course, as an evolution equation, Camassa-Holm contains infinitely many terms (each term is a differential polynomial in u).

  • Nikos Katzourakis added an answer:
    Does anybody know the minimal assumptions for uniqueness of strong solutions to the Dirichlet problem for linear non divergence elliptic systems?

    I mean to the Dirichlet problem for strong W2,p solutions to the linear elliptic system of the form

    Ααiβj (x) D2ijuβ(x)= fα(x)

    where A satisfies the Legendre-Hadamard and is (e.g.) continuous and f is Lp. There is a lot of material available in the literature for regularity estimates (e.g. the book http://link.springer.com/book/10.1007%2F978-88-7642-443-4 ) but I mean something for non-monotone (i.e. non-diagonal) systems in the spirit of the material of Sec. 9.1 in Gilbarg-Trudinger. Many thanks in advance!

    Nikos

    Nikos Katzourakis · University of Reading

    Thanks a lot, will check it!

    I just did, but the result refers to single equations, not strongly coupled systems.... 

    Thanks anyway!

  • Hussien Shafei 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.  

    Hussien Shafei · Faculty of science, SVU,Qena

    Dear Saravanan

    I read carefully your problem.

    First at all, your problem seems to be so easy, if you want to solve it Numerically. i.e. if all constants are given and the function fi(zeta) is explicit. then you can catch the values of the unknown values for a0, a1, d and b.

    Furthermore, If you want an explicit form for a1, you can let it to be a polynomial(as example) of suitable degree with unknown constants to be determined.

    Is my understanding for your problem is suitable?

  • Natalia S Duxbury added an answer:
    Do Maxwell's equations alone lead to constant velocity of light in all inertial frames?

    We normally think that the velocity of light is constant in all inertial frames is a postulate or the experimental result of Michelson-Morley experiment. But is it true that the Maxwell's equations lead to this and we do not need any extra postulate or any experimental results? Can anyone clarify this to me?

    Natalia S Duxbury · George Mason University

    Dr. Chatterji ,  this was an interesting question you posted !

  • Thomas I. Seidman added an answer:
    Is it possible to solve a non-linear second order PDE equation, containing one dependent variable and 4 independent variables?

    Is it possible to solve a non-linear second order PDE equation, containing one dependent variable and 4 independent variables.The dependent variable is the function and the 4 independent variables are x,y,z and t. where x,y,z denotes the x-direction, y-direction and z-direction in the space.Kindly suggest me some references and texts. 

    Thomas I. Seidman · University of Maryland, Baltimore County

    You have not mentioned the boundary conditions or the structure of the equation under consideration.  Typically, as Christoph said, efforts in this area seek either to prove well-posedness (in various settings) for boundary problems of a variety of structures (e.g., corresponding physically to reaction/diffusion or to eleasticity or to vibration or ...) or, assuming this, work with methods of computational approximation (e.g., by Finite Differences or series or Finite Element Methods or ...).  It all depends on what your final goal might be for the project.

  • Afshan Kanwal asked a question:
    How to solve a pantograph equation on maple?

    I am working on wavelets. I want to solve pantograph equation by using legendre polynomial.

  • Imran Khan Yousufzai 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)?

    Imran Khan Yousufzai · Mohammad Ali Jinnah University

    thank you Alessandro Pisano

  • Amaechi J. Anyaegbunam added an answer:
    What are the boundary condition for isotropic consolidation in triaxial cell for Biot's partial differential equation in 3D Space?

    In isotopic consolidation under triaxial test without radial drainage, flow in the sample in 1D but deformation is axisymmetric ie 3D. so how we will define the initial condition and boundary condition in X, Y and Z direction to solve Biot's partial differential equations of consolidation in 3D space? 

    Amaechi J. Anyaegbunam · University of Nigeria

    Dear Anurag Chafale

       Most problems in engineering are solved in an approximate fashion. The isotropic triaxial consolidation of a sample can be considered an axisymmetric problem. drainage is allowed from the sample bottom. The boundary conditions are that at the radius of the cell and at the top the drainage velocity is zero. I.e. at r = a, du/dt = 0. At the top say at z = 0, du/dt = 0 where u = pore pressure. At the bottom say at z = H, u = 0, i.e the excess pore pressure u = 0. Specifying the boundary values for stresses and displacements is more difficult because of the presence of the end cap for the triaxial cell.  

  • Ciprian G. Gal added an answer:
    Any Reference containing existence of weak(variational) solution of fractional (laplacian) poisson's equation of following type?

    I would like to get a reference containing the existence result of Fractional laplacian Poisson type equation, where the boundary of the domain is lipschitz . To be precise I added the question in pdf format. Please look after it .

    Ciprian G. Gal · Florida International University

    For the Holder regularity, see X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl. (9) 101 (2014), 275-302. This is the best result known presently.

  • Josef Schröttle added an answer:
    Can anyone help with intuition behind Stokes operator?

    I know that the definition of the Stokes operator (which appears in the functional form of the Navier-Stokes equations) is 

    A = -PLΔ

    where Δ is the Laplacian, and PL is the Leray projector (which maps a velocity vector u onto its divergence-free part). I acknowledge that in the setting of periodic boundary conditions, that this operator is equivalent to the negative Laplacian. However, is there a way to think about how this can be applied to bounded domains with no-slip conditions, which may not be periodic?

    Furthermore, how is the domain of the Stokes operator, D(A), defined in general? I know that it can be defined in terms of the Fourier transform in the case of periodic boundary conditions, but again, is it possible to picture this for a general bounded domain?

    Josef Schröttle · German Aerospace Center (DLR)

    Hi Oliver,

    thank you for the question. There is an answer in the lecture notes from J. M. McDonough on CFD of incompressible flows from the University of Kentuck to your question. As I understand it the Leray projector creates a divergence free vector field by substracting the gradient of a potential field from the physical velocity field by using Helmholtz decomposition. For periodic boundary conditions this is shown as you wrote and the projection of the negative Laplacian forms the Stokes operator. McDonough continues on pp. 31 with bounded domains. The velocity field has to fullfill certain criteria than, e.g. two times differentiable. First this is shown for no-slip without fluxes. A more general solution is however possible. Using these concepts could be a way to simplify e.g. numerical codes.

    Best wishes

    Josef

  • Torsten Asselmeyer-Maluga added an answer:
    Does anyone has knowledge of the Dirac Cohomology?
    The Dirac cohomology from the Dirac operator in the equation of the same name has been rencently studied, but by mathematicians (in particular Vogan) as a tool to solve problems of representation theory of Lie groups. It is the analog of the De Rham cohomology that has an application for electromagnetism. I'm looking for an account that is accessible for a physicist having a good knowledge about differential geometry and topology, including homology and cohomology. I'm especially interested in the dual homology.
    Torsten Asselmeyer-Maluga · German Aerospace Center (DLR)

    Interesting question. Dirac cohomology like BRST cohomology is in mathematical terms a Lie algebra cohomology theory. I think you are looking for Lie group cohomology which is in some sense dual to Lie algebra cohomology.

    But if you really want to construct a homology theory then you have to formulate the theory as homotopy classes of the space into a spectrum (cohomology). This can be dually formulated via smash products to obtain the homology theory.

    But what you wrote above you need the Lie group cohomology.

  • Abedallah M Rababah added an answer:
    What is the most frequently used orthogonal polynomial over [-1,1]?
    [-1,1] is the classical finite interval on which ops are defined.
    Abedallah M Rababah · Jordan University of Science and Technology

    Thank you dear Professor @Kennedy, I asked a classical question in Approximation Theory; many Mathematicians think that an orthogonal polynomial set is more important than others, but at the end each orthogonal polynomials set was created to serve for a proper purpose.

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