Abstract
This paper is concerned with the numerical blow-up solutions of semi-linear heat equations, where the nonlinear terms are of power type functions, with zero Dirichlet boundary conditions. We use explicit linear and implicit Euler finite difference schemes with a special time-steps formula to compute the blow-up solutions, and to estimate the blow-up times for three numerical experiments. Moreover, we calculate the error bounds and the numerical order of convergence arise from using these methods. Finally, we carry out the numerical simulations to the discrete graphs obtained from using these methods to support the numerical results and to confirm some known blow-up properties for the studied problems.
Highlights
In the field of mathematical modeling, many real life excremental problems can be contracted in the form of partial differential equations in different fields [1]
In tables (1) and (2), we present the numerical results of problem one, using explicit Euler scheme with respect to and 2, respectively
In tables (3) and (4), we present the numerical results of problem one, using implicit Euler scheme with respect to and 2, respectively
Summary
In the field of mathematical modeling, many real life excremental problems can be contracted in the form of partial differential equations in different fields [1]. Some of these equations have semilinear or nonlinear behavior that makes obtaining the exact solution of its governing equation difficult or without exact solution. It is well known that semi-linear parabolic equations arise in many physical situations, where the diffusion and source terms have to be modeled. In some cases the solution of the semi-linear heat equation cannot be continued globally in time, the so called blow-up phenomena, and that due to the infinite growth of the nonlinear term (source term) describing the evolution process. We consider the numerical solutions of the zero Dirichlet problem of a semi linear heat equation:
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