Numerical Methods And Stability Analysis For Two Classes Of Variable-order Fractional Advection-diffusion Equations

Fractional differential equations have emerged with the deepening of human understanding and transformation of nature.The complexity of derivative orders has spawned the expansion of classical fractional differential equations,that is,variable-order fractional differential equations.Advection-diffusion phenomena widely exist in nature,many fields such as environmental science,energy development,fluid mechanics,and electronic science use this as a research model,and in order to more accurately describe the time-related anomalous diffusion phenomenon in complex systems,scholars began to study the variable-order time fractional advection-diffusion equation.The existence of variable-order fractional operators and more complex special function forms make it difficult to obtain analytical solutions for such equations,so a series of effective numerical solutions have emerged.In this paper,two types of variable-order fractional advection-diffusion equations are selected and solved using a minimal search method based on the reproducing kernel theory.The main research results and innovations are as follows:In the first chapter,the background and significance of variable-order fractional differential equations are introduced,and the purpose of this study is determined by analyzing the current research status at home and abroad.Besides,the preliminary knowledge required for this thesis is given and the main research contents are briefly summarized.In the second chapter,the numerical method of one-dimensional variable-order time fractional advection-diffusion equations is discussed.Through integration processing,a set of Legendre multiwavelet bases in ~2L[0,1]space is transformed into orthonormal bases in solution space,which improves the accuracy.Then use piecewise parabolic interpolations to approximate the fractional derivative,which effectively simplifies the equation.After that,a minimal search method is proposed and the fact that using this algorithm to solve equations can find the minimum value is proved,it is worth noting that the accuracy increases steadily with the increase of the fraction and the number of basis terms.The final numerical example illustrates the effectiveness and stability of the given method.In the third chapter,a class of variable-order time fractional Mobile-Immobile advection-diffusion equations is considered.Using the improved differential quadrature method based on cubic B-spline to approximate the derivative of the function plays a role in simplifying the variables.Then the considered equation is converted into equivalent operator equations through symbolic definitions,which is also solved by the minimal search method,and the convergence order and stability analysis are given.Two effective numerical examples verify the feasibility of the proposed method.