Research On Numerical Algorithm For Fractional Integral And Differential Equations

Fractional equations can model many practical problems in the natural science and engineering.For example,fractional integral equations can model the electromagnetic casting process.Fractional differential equations can characterize random walk models.Fractional differential equations are important mathematical tools for modeling some problems with genetic properties and memory as well as nonlocality,which have been widely used in science and engineering fields such as biology,mechanics,finance,and other fields of science and engineering.However,the non-locality of fractional differential operators makes it difficult to find exact solutions for most fractional differential equations,constructing efficient and accurate numerical solution methods is the focus and key of the research.Moreover,some fractional differential equations can be transformed into equivalent weakly singular Volterra integral equations of the second kind,such as Caputo’s fractional differential equations.Weakly singular Volterra integral equations are a special class of fractional Volterra integral equations.However,the kernel function of such equations has singularity,which makes it difficult to solve analytically,and as the number of dimensions increases,the computational complexity also increases.In addition,fractional diffusion equations are generated by some anomalous diffusion models of a class of fractional differential equations.For some complex diffusion processes,the diffusion index may vary with space and time,so the distribution order diffusion equation comes into being.And the non-locality of fractional differential operators makes the numerical discretization of fractional differential equations produce large dense coefficient matrices,which increases the difficulty of the study.Therefore,there is an urgent need to design effective and high-precision numerical methods to study these mathematical models.This dissertation mainly studies the numerical solutions of the second kind of multidimensional weakly singular Volterra integral equations and two classes of distributedorder fractional diffusion equations.The main work is as follows,1.The numerical method for multi-dimensional linear weakly singular Volterra integral equations is proposed.First,the existence and uniqueness of the solution of the original equation are proved based on the generalized Gronwall’s inequality and mathematical induction.Secondly,the smooth transformation function is introduced to improve the regularity of the solution,avoiding the limitation that it is difficult to get a high-precision numerical solution of these equations affected by the low regularity of the solution.Then,the transformed equation can be solved by the multivariate Euler polynomial vector and the Gauss-Jacobi quadrature formula,which avoids calculating integrals and reduces the computational complexity.The numerical solutions of the original equation are then obtained using the smooth inverse transform function.The collectively compact convergence theory for the linear weakly singular Volterra integral equations is established by introducing a linear numerical integral operator,and the error estimation is analyzed in the framework of collectively compact convergence theory.The numerical results verify that the proposed method can obtain high-precision numerical solutions.2.The dissertation studies numerical methods for multi-dimensional nonlinear weakly singular Volterra integral equations.The multi-dimensional nonlinear weakly singular Volterra integral equations are discretized into corresponding algebraic equations by combining multivariate Euler polynomial vectors with the Gauss-Jacobi quadrature formula,which not only reduces the computational complexity,but also improves the computational accuracy.The system of discrete equations is solved by the modified Newton’s iterative method,which avoids the limitation of the traditional Newton iteration method by the initial condition and the regularity of the iteration matrix.Then,the numerical solutions of the original equations are obtained by the inverse transformation function.The collectively compact convergence theory for the nonlinear weakly singular Volterra integral equation is established based on the nonlinear numerical integral operator,and the existence and uniqueness of the approximate solution of the transformed equation as well as the error estimation are analyzed in the framework of collectively compact convergence theory.Further,the error estimation between the numerical solution and the exact solution of the original equation is derived by means of the smooth inverse transformation function.Finally,numerical results show that the proposed method is effective and easy to implement.3.The numerical method for a kind of two-sided space distributed-order fractional diffusion equations with variable coefficients is studied.Firstly,the spatial distribution order integral is discretized by the Gauss-Legendre quadrature formula,and the spatial fractional derivative is approximated by the weighted and shifted Grünwald-Letnikov operator.Then,the second-order accuracy approximation in space can be achieved.Secondly,the Crank-Nicolson method is used to achieve time discretization,and an unconditionally stable and second-order convergent difference scheme is derived.Furthermore,in order to avoid solving large systems of linear equations,an alternating direction implicit scheme is constructed by adding a small perturbation to the above second-order difference scheme.Finally,the stability and convergence of the two numerical schemes are analyzed.The numerical results are consistent with the theoretical analysis.4.The numerical method is studied for a class of time distributed-order and space variable-order fractional damped diffusion-wave equation.First of all,the variable-order fractional derivatives in the spatial direction are discretized using the weighted and shifted variable-order Grünwald-Letnikov operator.Then use the quadrature formula to discretize the distribution order integral in the time direction into the sum of multi-term Caputo fractional derivatives.Further,the Caputo fractional derivative is discretized using the weighted and shifted Grünwald-Letnikov operator.And it is proved theoretically that the numerical scheme is second-order convergent and unconditionally stable.Numerical results also show that the convergence order of the numerical scheme is very close to the theoretical value,and verify the reliability of the proposed difference scheme.