Finite Element Method and numerical study for a super-linear reaction-diffusion problem with integral conditions

doi:10.55059/ijm.2022.1.1/11

Abstract

In this work, we prove the existence, uniqueness, and continuous dependence of generalized solution of a nonlinear reaction-diffusion problem with only integral terms in the boundaries, by using the finite element method.Also we have developed an efficient numerical finite difference schemes. Some numerical results are reported to show the efficiency and accuracy of the scheme.

Highlights

Introduction and notationIn the recent years, a new attention has been given to reaction-diffusion problem which involve an integral over the spatial domain of a function of the desired solution on the boundary conditions; see [1 − 21]
A new attention has been given to reaction-diffusion problem which involve an integral over the spatial domain of a function of the desired solution on the boundary conditions; see [1 − 21]
The purpose of this paper is to prove the existence and uniqueness of a solution for the following non linear reaction diffusion problem with only integral conditions

Summary

Introduction and notation

A new attention has been given to reaction-diffusion problem which involve an integral over the spatial domain of a function of the desired solution on the boundary conditions; see [1 − 21]. The purpose of this paper is to prove the existence and uniqueness of a solution for the following non linear reaction diffusion problem with only integral conditions. X is the solution of the system (11) , if X is derivable and continuous function, for every each t ∈ I, X (t) ∈ I and X (t) = F (X (t)). We suppose that F is derivable continuous function on E ⊂ Rn. So for every each initial condition for t0 ∈ I and X0 ∈ E the solution of the system (11) if it exists it is unique. Let be t0 ∈ R and X0 ∈ Rn. If F is derivable continuous on X0, it exists h > 0 such that the solution of the system (11) verifying X (t0) = X0 exists on the interval [t0, t0 + h].

Statement of the problem

Construction of an approximate solution

A-priori estimates for approximations

Convergence and existence result

Numerical study with finite difference schemes

Numerical experiments