Application Of Quasi-interpolation In Numerical Solution Of Differential Equation

Burgers-Fisher equation is a very important fluid dynamic model and the study of this model has been considered by many authors. It has been found application in fields as gas dynamics, number theory, heat conduction, elasticity etc. Hyperbolic conservation laws is a classical topic in Computational Fluid Dynamics(CFD). As is known, the solutions of hyperbolic equation laws may develop discontinuities in finite time even when the initial condition is smooth. Recently, we found B quasi-interpolation is simple and effective for differential equations. In this paper, we study the application of cubic quasi-interpolation in differential equation. We present a new scheme for Burgers-Fisher equation and Hyperbolic conservation laws. The dissertation contains of four chapters.The first part introduces recent development of spline and B-spline, then generalized Burgers-Fisher equation and Hyperbolic Conservation Laws are introduced. These recent years, many researchers developed many numerical methods.The second part introduces the theory of B-spline, especially cubic B-spline quasi-interpolation. At last of this part, multi-spline is introduced for further study.Burgers-Fisher equation is studied in the third part by using cubic B-spline quasi-interpolation. The numerical scheme is presented, by using the derivative of the quasi-interpolation to approximate the spatial derivative of the dependent variable and a low order forward difference to approximate the time derivative of the dependent variable. At last, the numerical results obtained by this way have been compared with the exact solution to show efficiency of the method. Moreover, we study the stability of this method.We studied hyperbolic conservation laws in the fourth part by cubic B-spline quasi-interpolation, using the same method with the second chapter. The results are compared with the results which are obtained by MQ numerical method. Then, we verify our method for advection equation and one-dimensional Burgers'equation (without viscosity).The accuracy of BSQI may be not better than other methods, but the algorithm is very simple, so it is very easy to implement.