Solving 1D nonlinear PDE

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Firas Mourad 2016-12-19

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https://ww2.mathworks.cn/matlabcentral/answers/317436-solving-1d-nonlinear-pde

Hello to all,

I am trying to solve a nonlinear 1D (in space) PDE, and I am unsure how to set up my problem and which PDE solver to use.

The problem is as follows : (I will be denoting partial derivatives d(u(x,t))/dx by u_x)

u(x,t) and v(x,t) are two functions in time t and space x; t belongs to [0,T] and x belongs to [X1,X2].

My system is :

u_t = ( c1 * v * u^3 * (u+w)_x * abs((u+w)_x) + c2 * u^5 * ((u+w)_x)^3 )_x + e

v_t = c3 * ( exp(u-c4) - 1 )

Where c1, c2, c3 and c4 are known constants. w(x) and e(x) are known functions.

To make it more readable :

u_t = ( F[ u(x,t) , v(x,t) , w(x) ] )_x + e(x)

v_t = G[ u(x,t) ]

The boundary conditions are : u(X1,t) = u(X2,t) = 0

The initial condition is : u(x,0) = u0

As you can see it is kind of nasty looking and thus far I have failed to find helpful examples. I tried using the pdepe function, however, I get the following error :

Error using pdepe (line 293) Spatial discretization has failed. Discretization supports only parabolic and elliptic equations, with flux term involving spatial derivative.

I appreciate any kind of help.

Thanks in advance

Firas