Application of Legendre Wavelet Methods for the Numerical Solution of Fractional Differential Equations

This chapter focuses on the application of the two-dimensional Legendre wavelet technique for solving nonlinear fractional differential equations like the time-fractional Korteweg–de Vries (KdV)–Burger–Kuramoto (KBK) equation and time-fractional seventh-order KdV equation. Also, the time-fractional Kaup–Kupershmidt (KK) equation has been solved by using the two-dimensional Legendre multiwavelet method. The classical KBK equation is an important mathematical model arising in many different physical contexts to describe some physical processes in the motion of turbulence and other unstable process systems. The KdV type of equations are used to describe weakly nonlinear shallow water waves, have emerged as an important class of nonlinear evolution equation and are often used in practical applications. The chapter discusses the numerical solutions of fractional order partial differential equations (PDEs) comprising the Caputo fractional derivative. The advantage of Caputo&s;s approach is that the initial conditions for fractional differential equations with Caputo&s;s derivatives tackle the normal kind as for integer-order differential equations.