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Research On Several Kinds Of Numerical Methods For Fractional Convection-diffusion Equations

Posted on:2019-03-18Degree:DoctorType:Dissertation
Country:ChinaCandidate:X G ZhuFull Text:PDF
GTID:1360330623453363Subject:Mathematics
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During recent years,as the fractional calculus has been widely applied in mathematical modelling,fractional partial differential equations(PDEs)have attracted an increasing interest among academic circles,and played an important role in various fields of science as many as electronmagnetics,porous media mechanics,economical finance,environmental science,control theory,polymer material mechanics,and so forth.As a new class of mathematical models,these types of equations can provide a deeper and more accurate physical explanation for the complex dynamic behavior with memory effects,self-similarity and hereditary,and therefore possess incomparable superiority as compared to the traditional PDEs.Due to the limitation of analytic techniques,developing the related numerical methods has an important theoretical significance and the value of practical application.In this thesis,we investigate the numerical algorithms for the time-and space-fractional convection-diffusion equations,including the finite difference and collocation methods,operational matrix method and differential quadrature(DQ)method.The main contents are consisted of the following four parts:At first,a high-accurate numerical approach is discussed for the time-fractional diffusion equations with constant coefficients.Using a class of high-order difference schemes to decretize the Caputo derivative,and applying an exponential spline interpolation in space via a uniform nodal collocation strategy,a kind of finite difference-exponential B-spline collocation method is constructed.The unique solvability of the first-order scheme is analyzed and the unconditional stability is proved by employing a fractional von Neumann procedure.The numerical results confirm the theoretical analysis.Secondly,an operational matrix method is studied for the time-fractional convectiondiffusion equations with variable coefficients.Using Chebyshev cardinal functions as basis functions,an operational matrix is derived for Riemann-Liouville integration,where two kinds of means are performed to compute matrix entries.The advantages and disadvantages of these two generalizations are analyzed.The matrix approximations to spatial derivatives of the first-and second-order are further given.Based on an equivalent form with fractional integrations of governing equations,an efficient Chebyshev cardinal operational matrix method is proposed.The numerical results and comparisons with the existing algorithms illustrate its high accuracy and capability to handle the problems with a long time range.Thirdly,a spline-based DQ method is established to solve the one-and two-dimensional space-fractional convection-diffusion equations with variable coefficients.The differential quadrature formulations to the Riemann-Liouville derivatives are introduced,and the weighted coefficients are determined by using cubic B-splines as trial functions.Meanwhile,in order to rapidly compute the functional values of fractional derivatives at all discrete points,the explicit expressions of Riemann-Liouville derivatives of cubic B-splines are derived by a recursive formula of integration by parts.The resultant ordinary differential equations are tackled by the weighted average difference scheme.The method in presence inherits the principal features of traditional DQ methods,such as low computing cost,high accuracy,and the ease of programming.It can achieve the errors in the same scale of magnitude as finite element method(FEM)with far less CPU time.Finally,based on radial basis functions(RBFs),a DQ method is studied for the space-fractional diffusion equations with variable coefficients and the Caputo fractional directional derivatives on 2D irregular domains.The differential quadrature formulations to the fractional directional derivatives are introduced based on the functional values at scattered nodal points on the whole domain.The Multiquadric,Inverse Multiquadric,and Gaussian RBFs are utilized as trial functions to determine the weighted coefficients,and with them,a Crank-Nicolson type RBFs-based DQ method is developed,so is the algorithm flowchart,which is suitable for the space-fractional diffusion equations on arbitrary domains.The numerical examples cover square,trapezoidal,circular and L-shaped domain problems,and numerical results show its flexibility and the applicability to irregular domains.Due to the insensitivity to dimensional change,the proposed method can further be generalized to three-dimensions,and would not cause a significant increase in computing burden.
Keywords/Search Tags:Fractional convection-diffusion equations, Finite difference scheme, Collocation method, Operational matrix method, Differential quadrature method
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