# Partial Differential Equations

17
What is the difference between consistency, stability and convergence for the numerical treatment of any PDE?

For a numerical approach to any practical problems which are framed by Partial Differential Equations, we convert the PDE into any algebric equations with different schemes (implicit or explicit) like FTCS (Forward in time, Central in Space), LAX-Wendroff,  etc.

What is consistency, stability and convergence? And how these are tested and defined?

Please have a look on the following:

Abbott, M.B., and Basco, D.R., 1989, "Computational Fluid Dynamics, An Introduction for Engineers", Longman Scientific & Technical, Essex, England.

3.5. Consistency and convergence

3.7. Stability and accuracy

5
How can I solve eigenvalues and eigenvectors of fourth order ODE?

the equations is non-homogeneous, linear, fourth-order, partial differential equation governing the vertical deflection of the plate. you can see the governing eqution in photo below (eq.1). I would like to know "How to solve eigenvalues and eigenvectos of fourth order ODE ?"   A simple example will be helpful for me.

Dear Prof  Siriporn Anny,

Please see the following website which I think it is useful :

http://math.stackexchange.com/questions/1116652/eigenvalues-and-eigenfunctions-of-fourth-order-ode

--
Best Regards ,
Dr/Tarek Ibrahim ( T. F. Ibrahim )
Associate professor

¹Current address : Department of Mathematics, Faculty of Sciences and arts (S. A.) ,King Khalid University, Abha , Saudi Arabia

²Permanent address: Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt.
E-mails: tfibrahem@mans.edu.eg
tfoze@kku.edu.sa

Personal Websites :
http://eupc.mans.edu.eg/V2/get?Dr=27311031200455&T=2&L=A/
http://eupc.mans.edu.eg/V2/get?Dr=27311031200455&T=2&L=E
https://www.researchgate.net/profile/Tarek_Ibrahim2/?ev=hdr_xprf

34
How is a weak solution of a partial differential equation usefull in Physics and Engineering?

In the last few years I always thought as an engineer that the solution a physical system 'produces' is always smooth (differentiable to a certain degree). These solutions are so called classical solutions.

But now I have learned of weak solutions that can be found for partial differential equations. Those solutions don't have to be smooth at all, i.e. they have to be square integrable or their first derivative must be square integrable ...

So, if the weak solution is not differentiable it will not satisfy the original differential equation. Now, what is the use of the weak solutions that can be found? What is their physical meaning and how are they useful to find classical solutions?

I only would mention that there are the results about nonexistence of solutions as a natural way to a weak formulation of some partial differential equations ( see the papers from P.R.Garabedian from controlled plasma fusion  that are described by a variational principle based on magnetostatics ).

4
Is there any formal way to represent a nonlinear infinite dimensional systems ?

A formal representation of systems governed by ODEs is:

dot{x} = f(x,u)

y        = g(x)

Is there any formal way to define a nonlinear system with PDEs?

A good starting point is the excellent textbook "Partial Differential Equations" by L.C. Evans. Part III of the book is devoted to nonlinear pde's.

@book{evans2010partial,
title={Partial Differential Equations},
author={Evans, L.C.},
isbn={9780821849743},
lccn={2009044716},
year={2010},
publisher={American Mathematical Society}
}

47
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.

My Dearest Friend Zeev

You simply CANNOT resist the immense temptation to jump to conclusions on issues you hardly know at all ... !!! :-) :-) :-)

In principle, what you say is TRIVIAL : the domains of uniqueness of solutions MUST naturally depend on the respective PDEs.

The issue however is : HOW do they depend IN DETAIL ???

And for that, so sorry, so terribly sorry, you cannot avoid learning enough about the "order completion" method ...
And trust me, I am NOT going to give you here incremental lectures by answering to your - sorry to have to say - silly comments ... :-) :-) :-)

Yes, you should be more positive, polite, and above all HARD WORKING ... :-) :-) :-)

3
How can we make any ODEs or PDEs hybrid dynamic system?

How can we make any ODEs or PDEs hybrid dynamic system?​

thank you dear.....

4
What are dissipative and dispersive error for numerical treatment of PDEs?

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

hi you can found a good physical explanation of dissipative or dispersive error  and also on the amplitude error for the solution of Hyperbolic system in the  course notes by P. Colella and E.G. : UC Berkeley E266A.

+ 1 more attachment

2
Can somebody tell me how to solve, or give me the solution to this Partial Differential Equation: νz*(∂T/∂z)=α/r*(∂/∂r (r ∂T/∂r))+ μ*((∂/∂r)vz)^2?

I'm trying to find a temperature profile for a fluid and that's the equation i've found by now.

Thanks for your help. vz is the velocity of the fluid on z axe, r is the radius, a is a constant. I forgot to say that the fluid is moving through an annular space (the space among two cylinders, one inside the other) and it is given in cylindrical coordinates

7
Can the boundary layer problem be solved with parabolic approach of partial differential equation?

In general, air flow can be classified three kinds of method at the partial differential equation(PDE), and they are elliptic, parabolic, hyperbolic.

Parabolic is utlized for one-way flow such as wave propagation toward open domain (M=1).

Elliptic is utilized for commuicational flow, affect each other for all of the points, such as wave propagation in subsonic (M<1).

In case of boundary layer, marching to downstream until before separation, it is applied parabolic way in general.

I have a question at this point. Boundary layer(BL) almost occured under incompressible flow(M<0.3) and always under M<1. However BL theory almost be applied parabolic property, not elliptic despite of being under subsonic flow.

How is it possible?

Commonsensically,  based on the charater of BL, it is fair but theoretically hard to be understood to me.

Thanks for everyone answer my question, sincerely.

All of yours are beneficial informations.

Thanks.

12
How can solve the Partial differential equation?

How to solve this PDE either analytic solution or numerical solution?

give me roughly idea or method

I too would also like to recommend a most excellent, in-depth, and well explained book on PDEs.  Its title is:

Introduction to Partial Differential Equations, by P. J. Olver, (2013).

In this text, there are many examples in which PDEs are solved, both by the method of Characteristics and by Substitution, in the cases where it is possible to approach such problems via these techniques and procedures etc..  It is also very wide ranging in other areas, too, and is a 'must-have' PDE bible.

S.

3
Is there an analytical expression for the Green function of the 2D Klein-Gordon operator $\Delta u-k^2 u$ with Dirichlet conditions on the circle?

Is there an analytical expression for the Green function of the 2D Klein-Gordon operator $\Delta u-k^2 u$ with Dirichlet conditions on the circle?

I have found the answer to my question in the book of A.D. Polyanin: Handbook of linear partial differnetial equations for engineers and scientists, CRC Press, 2002. This is an excellent reference if you look for a concrete Green function.

3
Can Pazy's Theorem 3.1.1 be extended to the case of nonlinear semigroups?

In Pazy's book"Semigroup of Linear Operators and Applications to Partial Differential Equations ",  the Theorem 3.1.1: Let $X$ be a Banach space and $A$ be the infinitesimal generator of a $C_0$ semigroup $T(t)$ on $X$, satisfying $\|T(t)\|\leq Me^{wt}$. If $B$ is a bounded linear operator on $X$ then $A+B$ is infinitesimal generator of a $C_0$ semigroup $S(t)$ on $X$, satisfying $\|S(t)\|\leq Me^{(w+M\|B\|)t}$.

Now, if  $A$ be the  generator of a nonlinear  semigroup $T(t)$ on $X$,  can we still have the same result?  and why? Thanks.

3
Does anyone know about the homotopy analysis method in detail?

Dear researchers

I can not understand what happen in homotopy analysis method when it used for PDEs. could you help me by simple examples in details, particularly to show how R_(m-1) is calculated and how to plot the solution?

Dear Arash,

I have been reading papers on it too. However I would like to see an illustration on how the respective numerical algorithm is implemented through various steps to solve a given DE. I do not like the pre-designed matlab/mathematica packages which most of the times are black-box.

7
Anyone familiar with PDE theory on domains of infinite dimensional Banach spaces?

Does exist a theory for partial differential equations

on bounded domains or manifold of a generic Banach space (or Hilbert space)?

For example, consider G a bounded domain of a banach space E

and consider the problem

1. -D2u(x) = f(x), x in G
2. u(x) = 0, x in \partial G

where f in C0,a(G). If E is an n-euclidean space I know there is a

existence and uniqueness result for this problem.

In the case that E is a infinite dimensional Banach space,

what we already know about that problem? Is there a theory for it?

See for instance Gda Prato and JZabczyk,Second Order Partial Differential  Equations in Hilbert Spaces ,vol293-London Math Soc  Lect Notes ,Cambridge,UK,2002 or my articles ,OR SPECIALLY MY ORIGINAL STUDY 6.4 ON THE WEAK POISSON PROBLEM IN INFINITE DIMENSIONS -P183 WHICH HAS APPEARED ON MY RESEARCH MONOGRAPH  http://www.worldscientific.com/worldscibooks/10.1142/6856

6
In Crank-Nicolson method of solving one dimensional heat equation, what can be the maximum value of r (=k/h^2; k = time step, h = space step)?

I was trying to solve an one dimensional time dependent partial differential equation (similar to that of one dimensional heat equation) using Crank-Nicolson method. It would be very helpful if someone can tell me what could be the maximum value of r (=k/h^2; k = time step, h = space step) without affecting the credibility of the solution.

If you are dealing with simple heat equation, using Crank Nicolson scheme. The scheme will be unconditionally stable according to von-Neumann stability analysis,i.e the scheme will be stable for all values of r, but i need  to point out one point that you have to take care of the accuracy issue, means that we should choose h and k small enough in order to get highly accurate results. this will be very important if you are dealing with Crank -Nicolson and fourth order approximation for the space derivatives.

best wishes

5
Is it possible to solve a heat equation with Laplace Beltrami condition in 2D? How approximate the Laplace-Beltrami operator ?

Any suggestion by using the finite element method?

Look at the next tutorial: http://perception.inrialpes.fr/~Horaud/Talks/ECCV10-Tutorial4-Horaud.pdf

40
What are the advantages of numerical method over analyatical method?
We use several numerical methods. Why do we use it and is it really applicable?

11
What are the main drawbacks of traditional approaches to solving partial differential equations?

Hello, What are the main drawbacks of traditional approaches (e.g., finite element method, finite difference method, finite volume method, etc) to solving partial differential equations? Can they find solutions of any partial differential equations with 100% probability? Are they computationally expensive? Thanks!

Dear George,

Thanks for your suggestion and I've found Elemer Rosinger's works on his RG page.

Best regards, Xin

2
How might one apply differential transformation method in BBME (benjamin bona mahony equation)?

Can any body help me to check the transform equation.?

Dear Mohamed :  The link above is a good description of this method. And you may solve online the differential equation of interest . THe tool : Maxima software by William Schelter. The yamwi (yet another maxima web interface) is here :

http://maxima.cesga.es/

In order to apply the method you must know the way a recurrence equation is solved by Maxima. See the manual online here : http://maxima.sourceforge.net/docs/manual/de/maxima_70.html

18
Is their any numerical solution for 3rd order partial differential equations?

I was working on a simulation of Heat transfer block process that contain (liquid, steam and superheated steam). Unfortunately,  all the equations of the heat transfer model consist of a 3rd order P.D.E., I already know the finite element, finite volume and finite difference method. However, applying those methodology never worked with my situation here

So, I wonder if their is a more simpler way for numerical solving of P.D.E. I am currently using MATLAB r2010a to run the simulations

Thanks in advance for any contribution to this subject

Best Regards,

For a finite element modelling, You could try a better way using FreeFem++. A variational formulation need to be defined for the equation system ( heat transport, motion and continuity equation). Continuos and discontinuous polinomial can be considered. The implementation is based in your integral mathematical equations and the boundary conditions..

18
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.

I have used Mathematica to solve PDEs with no problems so far. I suggest you to use Mathematica with "NDsolve" solver and apply the Method of Line (MOL).

The MOL is an elegant semi-analytical approach that takes care about the stability of your solution specially the stiff problems.

8
How do I solve a system of partial differential equations?

I am stucked at a point:

I want to solve 3 Partial Differential Equations with 2 variables.

Thanks

Note: 1=v, and 2=Phi

Mr. Mittal is right except that Psi ^2 =cotv f'+C.

18
What methods exist about finding exact solution of nonlinear partial differential equations?

Any suggestion/resources are appreciated.

Thank you so much.

There is no exact solution to nonlinear pde.

3
Why are PDE's used to describe stochastic Calcium oscillation dynamics?

Why is that we used ODE's to describe deterministic calcium oscillation dynamics abut use PDE's to describe the stochastic calcium oscillation dynamics?

Thank you both @Manuel and @Bahram for clarifying my question.

5
How does one prove this Sobolev-type inequality in R3 ?

How does one derive the inequality

max {  | u(x) |:  x in R3  }   <=   ||  D u  ||_{L2}^{1/2}  ||  D2 u  ||_{L2}^{1/2}

for smooth, compactly supported functions  u : R3 --> R?

Dear Colleagues,

Thank you so much for kindly assisting me with the problem above, I trully appreciated your help. In particular, I would like to thank my dear friend and collaborator Prof. Lucas Oliveira for calling Xie's 1991 paper to my attention. There one finds a very nice derivation indeed (including the determination of the optimal value for the constant sitting in this inequality).

Warm regards,

Paulo Zingano

19
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.

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)

1
How to solve a pantograph equation on maple?

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

i am not expert in this field

3
How to solve the equation of beams on Pasternak nonlinear foundation?

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

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.

8
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.
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.

Hello Juan Ignacio Correa

In the expression [M]*C°+[K]*C=0, the partial derivative Co = ∂C/∂t  needs to be replaced by a finite difference approximation Co = ∂C/∂t = (Ct+δt-Ct)/δt and the other C with the matrix coefficient [K] needs to be expressed as C = θCt + (1-θ)Ct+δt where θ is a parameter between 0 and 1 that attempts to improve the accuracy of the approximation. Substituting these into your expression  [M]*C°+[K]*C=0 and rearranging gives you the complete equation. Taking θ = 1/2 gives the Crank-Nicholson approximation scheme. Taking θ = 0 gives an alternative fully implicit scheme. Please, try this out by yourself. Note that in my own formulation Co denotes nodal concentration at time t = 0. Wishing you success.