A decomposition method for a semilinear boundary value problem with a quadratic nonlinearity

Michael S. Gordon

doi:10.1155/ijmms.2005.855

Abstract

The author adapts the decomposition method of Adomian to find a series solution of a one-dimensional boundary value problem for a semilinear heat equation with a quadratic nonlinearity. Local and global convergence results are obtained.

Highlights

In this work, we consider the following semilinear boundary value problem for u(x,t) on the interval 0 < x < π, t > 0:∂tu = ∂xxu + γu2, u(0, t) = u(π, t) = 0, u(x,0) = f (x), where f ∈ C([0, π])
Our method arranges terms so that each linear problem is a PDE boundary value problem which is naturally solved with an expansion of eigenfunctions of ∂xx or a similar operator
With un defined by (4.3), (4.9), and (4.10), the series (4.2) converges uniformly on [0, π] × [0, δ] if f − a0 π < (1 − γa0δ)4/(4γδ) and f ∈ C([0, π])

Summary

Introduction

Adomian partitions (1.4) into a sequence of linear ODEs in either x or t whose solutions cannot generally be made to satisfy the boundary and initial conditions. Even when applied to an initial value problem with boundary conditions, the convergence of the solution depends sensitively on powers of f and its derivatives. In solving a similar problem in [1], Adomian must choose very specific initial data to guarantee local convergence in time.

Results

Conclusion