Is there a delta T or some other delta missing from this equation ?

hello, no this is a time  independant equation (poisson equation), delta u = constant *(1/(1+2*exp(1+u)))

Please look here for formulae: http://eqworld.ipmnet.ru/en/solutions/lpde/lpde302.pdf

For simple borders you can solve it even analytically. Numerical methods just to calculate the integral.

thank u but this is a non lineair equation it s not very simple to solve it analytically,

If you read the text, there is already a solution! Non-linearity is not in equation itself, but in RHS, logarithm is Green function, and then you just integrate RHS with it over space. Does it sound Chinese for you?

There is also reference to a book where this type of equations is called "linear": Babich, V. M., Kapilevich, M. B., Mikhlin, S. G., et al., Linear Equations of Mathematical Physics [in Russian], Nauka, Moscow, 1964

To satisfy boundary conditions may be tricky, but it depends on surface shape. The text gives analytical solution for a rectangular and circle.

Hi,

That answer from Dr Yegorov should be the best way.

I saw this question, with a matlab function, if other equations 

https://www.researchgate.net/post/How_to_solve_a_differential_equation_with_non-constant_coefficients

yes i understand what u talking about Yegorov, it seems for me clear from the text,,, but I am asked to look for a numerical solution and not analytic, my work consists in implementing the solution of this equation by the finite element method, and I have thought of newton raphson method or fixed point, thank u any way for your help ,

You can Taylor expand it to get exact solutions for small u.

Is u a function in 3D space?

lena , u 2D not 3D

So basically you want an FEM approximation of Delta u = f(u) with homogeneous boundary conditions. Left hand side : ordinary FEM for Delta u gives A u, with A symmetric negative definite, u in R^N. Right hand side Mf(u), with M ordinary mass matrix. giving a non linear system Au=Mf(u). It is not certain that this has a solution. It would have one if f'(u)>0, which unfortunately is not the case here. You could use some iterative method like Newton to find the solution if there is one.

Hi,

Is it a plate or circle? There are many solutions to nonlinear equations in general.

A book on FEM, heat conduction, and linearise around -2.

Jos J van Ka thank u for your answer, i use newton method but did n't work , there is something missing or a mistake in my code ..

@Arnaud Pacitti thank u for your answer, it's ok for the variationel form but i didn't understand the iteration process , du is the derivation of u or what? the increment ?? it s not really clear for me ,

thank u very much @arnaud pacitti, i'm a trainee at ifsttar, (LTN-versailles), for me this is the first time I deal with a nonlinear problem, i have already done an iterative method  but I think it is not correct,  .

Hello, i recomand you my paper:

https://www.researchgate.net/publication/268632469_Self-consistent_approach_to_solving_the_1D_Thomas-Fermi_equation_using_an_exponential_basis_set.

where we assess a non linear problem with freefem++ and an other spectral method using matlab. In fact, we use an iterative method to get the solution. the process is well explained.

I advise you to look for (picard method : the easy way or newton raphson method: with some constraints )

Conference Paper Self-consistent approach to solving the 1D Thomas-Fermi equa...

@arnaud pacitti, this is what u propose to me, for example 

mesh Th= square(10,10);

//Affichage et enregistrement

plot(Th,wait=1);

//********************************//

fespace Vh (Th,P1);

Vh u,v,delta,V=1/(1+exp(1+u)),C=exp(1+u)*(V^2); // du =delta u

//stop test newton

real eps=1e-6;

//initial guess of BC

u=0;

//

while(true) {

int k ;

real err =0; //initial erreur

// loop of newton method

for (k=0; k

ok , thank u @Arnaud Pacitti ,

Ps : i didn't find your phone number at ifsttar,

Just for kicks I did the following iteration

Au_(n+1)=Mf(u_n)

u_0=0

on a 20x20 grid with A linear fem laplace approximation, M lumped mass matrix, f(u)= 1/2+exp(u)

It converged with a convergence factor of 0.3. No need to use Newton

Jos J van Kan can u put the code u made please, thank u

Well, yes, no problem, but it's in APL. So I doubt whether the APL symbols would get across. But I'ĺl do it in pseudo code (/* ... */ are comments)

/*Preparation*/

Form the matrix A by linear elements.

/*That's a 400x400 block tridiagonal matrix with 20x20 size blocks*/

Initialize u=0 (that's a 400 vector)

Form the matrix M by lumping (that's a 400x400 diagonal matrix)

Set the accuracy you want to obtain, like tol=1E-10

Set residual = large (like 1E6)

/*end preparation *?

While residual > tol

Solve w from Aw=MF(u)

/* By any means, but one step block Jacobi would be adequate. That would be easy to program because it only requires to solve tridiagonal systems. If you want to be fancy, do one step block Gauss-Seidel. Converges at twice the speed.*/

Put u=w

Calculate R=MF(u)-Au

Set residual=max(abs(R))

end while