These are classic Navier Stokes equations and heat transfer or energy equation. I do not believe these are of 3rd order, however they are highly nonlinear and extremely difficult to solve with your own codes. It is recommended to use already available commercial software like ANSYS, COMSOL, or ABAQUS to solve such type of problems as they have the required solvers for such problems. A large number of example problems are also available with these software. They are mostly finite element or finite volume based software.

P.S. MATLAB have a very elementary PDE toolbox for solving PDEs. These equations can not be solved with that. 

Dear Prof. Mohamed,

Kindly accept my deepest appreciation for your reply,  the two types of equations that I am looking forward to solve are attached below, First the Navier Stock momentum equations,  and second the equation number (5.20) of Energy/Mass conservation

Kindest regards,

These are classic Navier Stokes equations and heat transfer or energy equation. I do not believe these are of 3rd order, however they are highly nonlinear and extremely difficult to solve with your own codes. It is recommended to use already available commercial software like ANSYS, COMSOL, or ABAQUS to solve such type of problems as they have the required solvers for such problems. A large number of example problems are also available with these software. They are mostly finite element or finite volume based software.

P.S. MATLAB have a very elementary PDE toolbox for solving PDEs. These equations can not be solved with that. 

I think radial basis function collocation method can be an effective alternative for 3rd order PDEs.

Dear you can use the discontinuous Galerkin finite element method, the method use the Legendre polynomial as a weak function 

Dear Prof. Khaled Rady, as an alternative you may use FLUENT sotware from ANSYS, as Prof. Singh said...

I think that the problem geometry and boundary conditions can be very significant here.The commercial programs that the former colleagues have indicated are probably the most practical option. But if you have no access to them, for certain simple geometries there may be other methods that can be tested with the help of Matlab.

Since your equation has time component, I am assuming you are solving a dynamic problem. These are direct numerical solution and have been performed in different geometries. For simple goeometries, such as Couette flows, this is very much solvable using finite difference method and ensuring the deltat and deltax satisfy the stability condition. If you are trying to simulate turbulence (i.e., large Reynolds Number), then you would need to look at some of the articles on Direct Numerical Solution of Turbulent Flows. If your heat equation see

Dear Mr. Rhady,

From my understanding, also following the previous comment of Mr. Singh, I do not believe that the classic Navier-Stokes equations are PDEs of third order, but second order. The highest derivative lies within the viscous terms of the momentum equations. Could you state why you think the PDEs are of third order - Maybe there is a missunderstanding from my side here.

You can use finite differences for the Couette and Poiseuille flow.

Radial basis funcrions such as mq, and Gaussian can provide much higher convergence rate than fem, and fdm in some cases. But you need to deal with ill-conditioned system.

 There equations are called the Navier-Stokes equations. They are non-linear PDEs of second order, not third order. Since you are modelling heat transfer, besides the three momentum equations in x, y and z directions you must consider an extra equation for a scalar (temperature, if that is the case), assuming you are modelling a incompressible flow.

Actually, you can solve these equations in either finite element, finite difference or finite volume methods. There are plenty of softwares available with this methods already implemented, including commercial and non-commercial. If you are interested in free and open-source software, I would suggest you to try OpenFoam, a great library written in C++ which solves a great variety of physical problems (including solid mechanics, fluid dynamics, heat transfer, electromagnetism and so on). It is based on the finite volume method.

Dear all prof. and Eng.,

Kindly, accept my deepest gratitude and appreciation for spending your precious time in answering my question and I hope you all are treated with the same generous manners for any of your inquiry.

I am so impressed that I was finally able to solve the Navier Stocks P.D.E using comsol with the linkage to Matlab, I can't imagine how far can any mechanical design be improved when using evolutionary computing( genatic alogarithm) and online testing the model performance using Navier stocks without having the perposed prototype ...!

Best regards,

Khaled Rady

That is good for you

I agree with Prof Singh, especially concerning COMSOL.  You may wish to know that this Finite Element software was once a tool box in Matlab called FEMlab.  Now it is much better and expanded and can handle multiple phases, such as your steam and liquid water; however, the physics of interaction (boiling or condensing) is non-trivial and requires special consideration.  Dr. Kurt Lund, COMSOL Consultant.

You could try the Open Source software called OpenFOAM for your heat transfer problem. Since you have steam and air, you would be having flow as well. So as someone above points out, you would have to deal with heat and mass transfer together. Navier Stokes equations with heat transfer could be considered as a model for such a situation. Such equations and their modifications which you might have, can be implemented without too much difficulty in OpenFOAM. You will have to spend some time to familiarize yourself with the data structures used in OpenFOAM and its object oriented programming style.

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..

Dear Khaled Mohamed Rady ,

    I'm a PhD student working in nonlinear dynamics. I've also with the same situation of solving the following equation,

     Q_{t} + Q_{xxx} = 3QV_{x} + 3VQ_{x}.

  As it has third order term and its a non-linear equation I cannot able to solve it with finite-difference scheme. Could you direct me how to solve this kind of equation numerically ?

Thanks in advance.

Dear sudharsan,

You can try Comsol 5 partial differential equation solver, I have attached a link for the comsol tutorial video

Best Regards,

Khaled Rady

https://www.google.com/url?sa=t&source=web&rct=j&url=http://m.youtube.com/watch%3Fv%3D9yw4IvqCv1g&ved=0ahUKEwiV7ouAqYXNAhXHMBoKHeaHD-cQtwIIGjAA&usg=AFQjCNF4iBQMA0ivYd2U4PZlCU8yEW_84A&sig2=PstiSiEXZLyGMRwHMhBZdA

https://www.google.com/url?sa=t&source=web&rct=j&url=http://www.comsol.com/community/forums/general/thread/15007/&ved=0ahUKEwiV7ouAqYXNAhXHMBoKHeaHD-cQFggmMAQ&usg=AFQjCNF1ZENsdOC8zbjTeObR65CbStBC3Q&sig2=-jFZZMS-8n84XtAKT4C7xw

www.csc.kth.se/utbildning/kth/...cl08/labbar/IntroPDEFEMEng.pdf