# Partial Differential Equations

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

2
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.

In the case when A is a maximal monotone operators check out: Brezis, H: Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Mathematics studies 5, North-Holland Publishing Co., (1973).

• Gabriel Caicedo asked a question:
Open
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.

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.

4
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.

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

8
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.

3
What is the name of this inequality?

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

Thank you

16
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

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.

8
Is Fractional Calculus useful in every branch of science?
I.e. Applications of fractional calculus.

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.

• Source
##### Article: Fractional Derivative of the Hurwitz ζ-Function and Chaotic Decay to Zero
[Hide abstract]
ABSTRACT: In this paper the fractional order derivative of a Dirichlet series, Hurwitz zeta function and Riemann zeta function have been explicitly computed by using the Caputo fractional derivative in the Ortigueira sense. It has been shown that the obtained results are a natural generalization of the integer order derivative. Some interesting properties of the fractional derivative of the Riemann zeta function have been also investigated to show that there is a chaotic decay to zero (in the Gaussian plane) and a promising expression as a complex power series.
Journal of King Saud University - Science 04/2015; 10. DOI:10.1016/j.jksus.2015.04.003
4
What is the meaning of the Laplace operator whose one eigenvalue is 1?

This is related to global analysis.

who will send you, profesor sergeyyy :)

8
Do you have solved examples of system of nonlinear pde using finite element method?

See above

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.

13
Is there any numerical solution to solve nonlinear coupled PDEs?

Is there any numerical solution to solve nonlinear coupled PDEs?

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

5
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?

You can use Mathematica program