Quasi Wavelet For The Space-variable Order Fractional Partial Differential Equation

The fractional operators have a long history. For three centuries the the-ory of fractional derivatives developed mainly as a pure theoretical field of mathematics useful only for mathematicians. However, in the last few decades many authors find fractional Calculus are used to mechanics, optics, biology and so on.With the problem complex, many authors discover the exponent of fractional partial differential changes with time and space in many dynamic process. So the variable-order fractional differential partial equations ap-pear. However, the variable-order fractional partial differential equations have variable-order exponents. It is difficult to study its analytic solutions. Besides,the research on variable-order fractional partial differential equations is rela-tively new, and numerical approximation of these equations is still at an early stage of development. At present,there are some articles as following, such as Chen, Liu, Anh.et al [3,4,5,6,7]; Lin, Liu, Anh.[8], Zhuang, Liu, Anh.[9], Shen. shujun[13], Coirnbra[14], Sun, Chen W,Chen Y[15], Soon, Coimbra, Kobayashi[16]. But these authors use difference, interpolation, RungeKutta and Crank-Nicholson to study the numerical solution for variable-order frac-tional partial differential equations. We never find quasi-wavelet algorithm for numerical approximation of these equations. Hence,it is meaningful to use quasi-wavelet method for a variable-order fractional advection-diffusion equation.In this paper, we mainly discuss the quasi-wavelet algorithm for a variable-order fractional advection-diffusion equation with a nonlinear source term. At the same time,we use numerical examples to test the quasi-wavelet algorithm of the validity and accuracy for this problem. The Euler method is used to discrete time and the two methods are used to discrete space. One is simple quasi-wavelet method; the other is double quasi-wavelet method. In this article, main results follows:(1)Given the main introduction about variable-order fractional derivative, the development of variable-order fractional partial differential equation and the research on numerical approximation of these equations.(2) Given the main introduction about quasi-wavelet method.(3) Given the time semi-discrete form and space-time fully-discrete form.(4) Given the numerical experiments result. Test the Euler-quasi-wavelet algorithm for a variable-order fractional advection-diffusion equation with a nonlinear source term of the validity and accuracy.