Thanks Peter for the quick response. 

Actually I already tried dv/dt=a part but I missed the dv/dx=f part. Thanks a lot for pointing out. Let me try and I hope it works.

You have many numerical techniques to solve this coupled system , can't you 

Peter has already told you a lot..use repeated substitutions and get the system reduced to a set of first order ODE's and then apply techniques for linear systems

You can use Mathematica-Wolfrom program

It is very simple and effective to solve this type of equations.

The first equation describes the vibrations of a beam. Depending on F, there are exact solutions based on Fourier series, The second equation could be of a string if w=v. Again, this can be solved exactly.

@Sudhakar Nair Sir the issue is that both equations are coupled so that means we need to solve both equations simultaneously ... 

Well you can try bvp4cin matlab after reduction of the coupled equations to system of first order ordinary differential equations. or i will suggest you use the numerical method  called SHAM.

i can put you through if you inbox me on [email protected]

Thanks Sir @Adeyemi ... I have sent an email to your inbox. I will be waiting for your response. 

Dear you can also use Comsol. it is solve many problems type same as PDEs. ODEs. ...etc

the results by that is program obtain by finite element method

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.

Do not know of analytical method, however can be solved numerically. Define a new variable P = d2v/dx2. Substitute this in first two equations, 4th order is eliminated and you get 2nd order PDE's with additional equation P = d2v/dx2. Solve these three coupled PDE's for v,w, and P.

Hi Peter,

I think the description of the problem is rather incomplete. It is a one-dimensional PDE so we need to define a x-interval over which problem has to be solved. We also need Boundary conditions for variables v and w as well initial conditions since it is a transient problem. The solution also depends on what is F? If this much information is provided I believe the problem can be solved numerically. 

Hi Peter and Ashish 

First of all thanks for your replies. 

Secondly Let me clarify the problem description in the light of your comments.

1. In the problem above x and t are two independent variable and are not related among themselves. 0

Hello Muhammad,

If you know the BC's and IC's use COMSOL or any other software which can solve coupled PDEs. In COMSOL it is very straightforward, select a 1D Coefficient form PDE with three dependent variables v,w, and P. Write the three equations in specified format, supply BC's and IC's information, and solve using a time dependent solver. 

Hello Muhammad Usman

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)