Dear Simran, if the equations are highly (what do you mean with this word?) non-linear, probably no analytical method exists for exact resolution. Regarding numerical methods, the best are Runge-Kutta (are time derivatives explicit?) and their specialized versions.  Gianluca

Even with an analytical, it is impossible to obtain a numerical value in several cases.

the example are of Hadamard for example,

The problem is not easy. Nonlinearity and the time may generate the troubles. If the variety in time are harmonic I would suggest solution in frequency domain (the first harmonic method could be applied to take into account nonlinearity).

But generally (I do not know lot about the problem) I could suggest (coupled problem) COMSOL, which would be able to solve such a problem (nonlinear, couplet and time dependant).

Regards

J.S.

Thanks sir, i tried the solution by LAPLACE DIFFERENTIAL TRANSFORM METHOD (semi analytical method) and whereby time domain is solved by laplace and space domain by differential transform method. Even, I am able to get recursive relation after formulation. Now, I stuck in solving nonlinear implicit recursive relations.

Well to me the following methods are better:

Numerical method (unsteady) - Explicit or implicit finite difference method

Numerical method (steady) - Nactsheim-Swigert shooting iteration technique together with Runge-Kutta six order iteration scheme

Semi-analytical - Homotopy analysis method

Analytical - Method of Perturbation

At the monent,  Adomian decomposition method is the best and  simple to implement , over all methods. This is because it gives both analytical and numerical solutions way beyond every other method.

My best guess would be to uncouple the two pde's by introducing an additional variable w which act as a control in each pde with respective solutions u(w) and v(w). Then add the constraint u(w) = v(w) and solve an optimal control problem to minimize the (squared) norm ot the error (u-v). For an efficient numerical solution use an augmented lagrangian and the celabrate ADMM algorithm (I introduced back in 1976!). It will work particularly well if you have fast solvers for the the two now independent pde's.

To make the decomposition argument clearer, the auxiliary control variable take the place of v in the first pde and of u in the second pde. Actually a better regularized optimization problem is obtained if you introduce the constraints u(w) = v(w) = w and minimize the sum of the (squared) norms of the errors u-w and v-w.

It depends what kind of expertise you have. If you are good enough in programing a lots of good Libraries /software are available to solve such kind of coupled PDEs via Finite element methods like DEAL.II , Dune PDEs numerics and many more . It should be all related with numerical solution.These are helpful for complex geometry problems as well.

Sir Taj Munir,

Can you please name all that softwares and solver which are capable of solving such PDE,s.

Thanks in advance.

Daniel Gabay Sir,

Can you please provide some solved examples for the method you have explained.

Thanks in advance 

I wrote  these in my first suggested answer.

Dear Simran,

I have spent 30 years looking for analytical solutions for some coupled nonlinear differential equations that are probably simpler than the ones you are looking at. I made progress but still have not found answers to all of the questions. In terms of numerical methods, very sophisticated software have been developed for specific applications but a different software package is better for different applications. Even with the best software package for the application, convergence to a numerical solution is not guaranteed. You might get lucky in that your very specific problem has a simple analytical solution. But if your goal is to solve a more generic problem, then I would say that there is no easy answer. 

It is possible to try Group Analysis to construct analytical Solutions. Other methods which can be used are also Painleve-analysis and also First-Integral Method. The last one can be applied after transforming the System of partial differential equations by using so called travelling wave variables to an ordinary differential equation.

@Maken

(1) Use MATLAB  to get the answer .

(2) Then find analytical one .

First try vander pol oscillator . Then switch to your problem . Hope it will be helpful.

B.Rath

I prefer to convert them or transform to Z transformation and find out analytical solution.

If it is difficult, I prefer to use numerical techniques such as a finite higher order discretization methods to solve the problem

How about making the question a little more specific so that the answer will solve one of my problems. The equation is

P'(x)=A*(P(x)+N) / (P(x)+N/2) - G(x)*(P(x)+B) / (P(x)+N/2)

where P(x) is the unknown to be solved in terms of a specified (non-negative) endpoint value P(0), A is a constant, B and N are positive constants (with B < N), and G is an increasing function of x that satisfies G(0) = 0 but is otherwise arbitrary. Does anybody know how to get an analytical (not numerical) solution for this? To be useful it must apply to arbitrary G satisfying the stated conditions.

You can use the Analog Equation Method (AEM) to solve efficiently coupled partial differential equations. The method is in detail explained in the book "The Boundary Element Method for Plate Analysis" by J.T. Katsikadelis, Academic Press, Elsevier, 2014. Several examples problems described by coupled PDEs are presented in Section 3.5, 4.2 and Section 5.

https://www.wiley.com/en-us/Advanced+Numerical+and+Semi+Analytical+Methods+for+Differential+Equations-p-9781119423423

Actually this depends on various things: one is that what type of solutions in which are interested exact type or Numerical type;. and second one that what type of coupled system; linear or nonlinear and BVP (Boundary value problem) or IVP (Initial value problem) or problem without any condition.

Moreover there are various methods which reduce the number of independent variables in the case of PDEs e.g Lie symmetry analysis

The second thing is that you can utilize classification approach

Moreover you can use any numerical approach if that particular approach fulfill your underlying conditions.

It actually depends on type of problems considered. In a specific type of problem, researchers developed methods accordingly. In general. we can see in the literature that new methods have been proposed from time to time. And this involve newly introduced method, improvement of the previous method and combining/hybridizing existing methods. That is why in most paper discussing on methods, it is always important to perform comparison in term of errors produced.

I think Dr if nonlinear PDE variational iteration method easy to contract no require to decompose the nonlinear terms

حوت دكتور علي ما شاء الله.