Abstract
A numerical method based on finite difference scheme with uniform mesh is presented for solving singularly perturbed two-point boundary value problems of 1D reaction-diffusion equations. First, the derivatives of the given differential equation is replaced by the finite difference approximations and then, solved by using fourth order compact finite difference method by taking uniform mesh. To demonstrate the efficiency of the method, numerical illustrations have been given. Graphs are also depicted in support of the numerical results. Both the theoretical and computational rate of convergence of the method have been examined and found to be in agreement. As it can be observed from the numerical results presented in tables and graphs, the present method approximates the exact solution very well.Keywords: Singular perturbation, Compact finite difference method, Reaction diffusion.
Highlights
Any differential equation in which the highest order derivative is multiplied by a small positive parameter (0 1) is called Singular Perturbation Problem and the parameter is known as the perturbation parameter
Differential Quadrature Method (DQM) is based on the approximation of the derivatives of the unknown functions involved in the differential equations at the mesh point of the solution domain and is an efficient discretization technique in solving boundary value problems using a considerably small number of grid points
Most of the existing classical finite difference methods which have been used in solving singular perturbation problems give good result only when the mesh size is much less than the perturbation parameter which is very costly and time consuming
Summary
Any differential equation in which the highest order derivative is multiplied by a small positive parameter (0 1) is called Singular Perturbation Problem and the parameter is known as the perturbation parameter These types of problems arise very frequently in diversified fields of applied mathematics and engineering, for instance fluid mechanics, elasticity, hydrodynamics, quantum mechanics, chemical-reactor theory, aerodynamics, plasma dynamics, rarefied-gas dynamics, oceanography, meteorology, modeling of semiconductor devices, diffraction theory and reaction-diffusion processes and many other allied areas. It is well-known fact that the solution of singular perturbation problems exhibits a multi-scale character; that is, there are thin transition layer(s) where the solution varies rapidly or jump suddenly known as boundary layer, while away from the layer(s) the solution behaves regularly and varies slowly known as outer region. Compact finite difference method is a finite difference method which employs a linear combination of three consecutive points of derivatives to approximate a linear combination of the same three consecutive values of a function y(x j ), j i 1, i, i 1
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
