The finite difference method on adaptive mesh for singularly perturbed nonlinear 1D reaction diffusion boundary value problems

Abstract

In this paper, we study singularly perturbed nonlinear reaction-diffusion equations. The asymptotic behavior of the solution is examined. The difference scheme which is accomplished by the method of integral identities with using of interpolation quadrature rules with weight functions and remainder term integral form is established on adaptive mesh. Uniform convergence and stability of the difference method are discussed in the discrete maximum norm. The discrete scheme shows that orders of convergent rates are close to 2. An algorithm is presented, and some problems are solved to validate the theoretical results.

A Numerical Method for Solving Singularly Perturbed Quasilinear Boundary Value Problems on Shishkin Mesh

“…Many different phonemena in science can be modeled by them. They emerge in electrochemistry [22], control theory [7], nuclear engineering [1], fluid dynamics [12] and plasma physics [6] (see, also the references therein).…”

“…Therefore, some important techniques have been introduced in the literature. These involve: finite difference methods [1,4,5,8,11,12,15], fitted mesh method [23], reproducing kernel method [7], factorization method [25], Galerkin method [18,26], differential transform method [9], variational iteration methods [2,10] and so on [6,16,19,20,27].…”

“…In recent years, some discretizations have been suggested for singularly perturbed quasilinear problems. In [11,12], nonlinear singularly perturbed reaction-diffusion type problems were discretized on layer-adapted meshes. The finite difference scheme with exponential coefficient on piecewise equidistant mesh is established for delay form of singularly perturbed quasilinear problems [13,14,17].…”

“…The novelty of this study is that the quasilinear term contains the derivative of the function u(x). Motivated by the works in [11,12], this research aims to present an efficient and new difference scheme for solving singularly perturbed quasilinear boundary value problems on S-mesh.…”