Abstract
In this paper, the boundary value inverse problem related to the generalized Burgers–Fisher and generalized Burgers–Huxley equations is solved numerically based on a spline approximation tool. B-splines with quasilinearization and Tikhonov regularization methods are used to obtain new numerical solutions to this problem. First, a quasilinearization method is used to linearize the equation in a specific time step. Then, a linear combination of B-splines is used to approximate the largest order of derivatives in the equation. By integrating from this linear combination, some approximations have been obtained for each of the functions and derivatives with respect to time and space. The boundary and additional conditions of the problem are also applied in these approximations. The Tikhonov regularization method is used to solve the system of linear equations using noisy data. Several numerical examples are provided to illustrate the accuracy and efficiency of the method.
Highlights
Most of the physical problems arising in various fields of physical science and engineering are modeled by nonlinear partial differential equations (NLPDEs) [1]
The generalized Burgers–Huxley and generalized Burgers–Fisher equations are of the form ut = εuxx − αuδux + βu 1 − uδ ηuδ − γ, a < x < b, t > 0, ð1Þ
The boundary value inverse problem related to the generalized Burgers–Fisher and generalized Burgers–Huxley equations was solved numerically
Summary
Most of the physical problems arising in various fields of physical science and engineering are modeled by nonlinear partial differential equations (NLPDEs) [1]. The generalized Burgers–Huxley equation describes a wide class of physical nonlinear phenomena, for instance, a prototype model for describing the interaction between reaction mechanisms, convection effects, and diffusion transports [8] It has found its applications in many fields such as biology, metallurgy, chemistry, combustion, mathematics, and engineering [8, 9]. For the first time, a boundary value inverse problem for the generalized Burgers–Huxley and generalized Burgers–Fisher equations will be studied. For this purpose, first, a quasilinearization method is used to linearize the equation in a specific time step.
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