High-precision And Efficient Algorithms For Fractional Differential Equations

Fractional differential equations are widely used in fluid mechanics,turbu-lence and viscoelastic mechanics,abnormal diffusion,fractal and dispersion in porous media,signal processing and system identification,electromagnetic waves and other fields.The non-locality of fractional operators provides a better explanation for materials with memory and genetic properties in the real world,which is more conducive to modeling various complex mechanics and physical behaviors.However,in most cases,fractional differential equations cannot be solved analytically.Only a few of them have analytical solutions expressed by means of complex functions such as Mittag-Leffler function,H-function and Wright function,and these functions are more difficult to calculate.Therefore,many scholars devote themselves to studying its numerical solution.Common numerical solution methods include finite difference method,finite element method and spectral method.In addition,there are a small number of solutions using finite volume element,and meshless methods.Actually,the fractional order is non-local,which is different from the traditional integer order derivative,so the numerical format for solving the fractional order equation usually requires a extremely large storage space and calculation amount.For this problem,we have proposed some fast solutions,such as:fast Fourier transform,sum-of-exponentials(SOE),proper orthogonal decomposition technology(POD),etc.However,there are relatively few researches on fast and efficient solutions to time fractional equations.In this thesis,we will study high-precision and efficient solutions to time fractional differential equations.In this paper,firstly,we establish a Crank-Nicolson type compact finite d-ifference scheme for one-dimensional and two-dimensional fractional Cattaneo equations with Caputo-Fabrizio derivatives,and theoretical analysis is conducted.In addition,because of the huge amount of computational work for the above format along with direct solution,we propose a quick solution method based on the recursive relationship between adjacent time layers of the numerical format,which effectively reduces the amount of calculation and storage.Secondly,considering the time distribution order partial differential equation based on the Caputo-Fabrizio derivative,two unconditionally stable formats are developed,and their convergence rates are O(?2+h2+??2)and O(?2+h4+??4),where ??,h and? represent the distribution order,space and time division steps,respectively.Thirdly,observing that the solutions of fractional equations usually have weak singularities,we solve the equivalent integral equations of fractional nonlinear differential equations and linear partial differential equations on three non-uniform grids,which are proposed according to the positive integer power summation formula,and the error estimation of the numerical format is performed.Fourthly,the solution of the time fractional diffusion equation with Caputo derivative is weakly singular.We use the L2-1? format to discretize the fractional derivative on three non-uniform grids.The stability analysis and error estimation of the numerical format are carried out.Fifthly,consider one-dimensional and two-dimensional time fractional diffusion equations with Caputo fractional derivatives.In order to avoid the deterioration of the numerical format caused by the solution on a uniform grid,we use the L1-2 format on the graded grid to discretize the time fractional derivative,and the local truncation error is estimated for the fractional derivative discrete format.In addition,a reduced-order extrapolation algorithm is proposed for the numerical format,which effectively reduces the amount of calculation.Sixthly,considering that the finite difference method and finite element method need to construct the grid in advance,which is not convenient to solve the complex area problem,we establish the finite difference/RBF meshless scheme for the two-dimensional time fractional convection diffusion equation with the Caputo fractional derivative.And use the RBF reduced-order extrapolation algorithm to reduce the amount of calculation.Specifically:In chapter 1,firstly,we give a brief introduction to the development of fractional calculus,and give several definitions of fractional derivative.Then,a brief introduction to the research content of this thesis is given.In Chapter 2,a fast and compact finite difference method is proposed for the Cattaneo equation with time fractional derivative without singular kernel.We first study the one-dimensional problem.The spatial derivative term is discretized by compact difference operator,and the Crank-Nicolson approximation is used for the Caputo-Fabrizio fractional derivative to establish the numerical discretization format of the Cattaneo equation.Then,the stability analysis and error estimation of the discrete scheme are carried out,which proves that the proposed compact finite difference scheme has fourth-order spatial accuracy and second-order time accuracy.Subsequently,we generalize the one-dimensional problem to the two-dimensional problem,derive the high-order format,and give the corresponding theoretical analysis.In addition,because the fractional derivative is relevant to history and non-local,it requires huge storage space and computational cost,which means extremely high consumption,especially for long-term simulations.We analyze the discrete format of the time derivative and find that there is a recursive relationship between the numerical formats of the adjacent time layers.Based on this,we give an effective and fast solution to the Caputo-Fabrizio fractional derivative,so that the amount of calculation is reduced from O(MN2)to O(MN),and the storage capacity is reduced from O(MN)to O(M).Finally;through some numerical experiments,the correctness of the theoretical analysis and the feasibility of the fast algorithm are verified.In Chapter 3,two effective finite difference schemes are developed for the time distribution order partial differential equations with Caputo-Fabrizio fractional derivatives in one-dimensional space.One uses a compound trapezoidal formula to approximate the integral term,and uses a second-order central difference quotient approximation for the spatial derivative term;The other uses a compound Simpson formula to approximate the integral term,and a compact difference operator for the spatial derivative.The stability analysis and error estimation of the above two formats prove that these two formats are unconditionally stable in the sense of discrete L2 norm,and their convergence rates are respectively O(?2+h2+??2)and O(?2+h4+??4),where ??,h and ? are the distribution order,space and time division steps,respectively.Finally,the theoretical results are verified by numerical examples.In Chapter 4,we discuss the fractional-order nonlinear ordinary differential equations with Caputo derivative and linear reaction-diffusion equations.Usually the solutions of fractional differential equations have weak singularities at the initial moments.If the finite difference method is used to solve them on a uniform grid,it is difficult to obtain the optimal convergence order for the resulting format.The article[1]converts the fractionalorder nonlinear ordinary differential equations into equivalent integral equations,and divides the time region non-uniformly.The integral terms are approximated by compound rectangular formulas and compound trapezoidal formulas,respectively.In addition,considering that the calculation of nonlinear equations using the above two methods is more complicated,the prediction correction format is introduced.Theoretical analysis and numerical experiments show that the regularity of the equation solution has an impact on the convergence order.We expand this idea.According to the k power formula,we propose two other non-uniform grids for k=4,5.The equivalent integral form of the fractional-order nonlinear ordinary differential equation is newly proposed.The above-mentioned method is used for discretization in the format,and theoretical analysis proves that the discretization problem on the newly proposed grid can provide a better convergence order.In addition,we consider a fractional-order linear reaction-diffusion equation with weak singular solutions and transform it into an equivalent integral equation,and use a compound trapezoidal formula to approximate the integral term on three non-uniform grids.The spatial derivative is discretized using the finite difference method on a uniform grid.The convergence analysis of the numerical format shows that we can obtain the optimal convergence order in different non-uniform grids.Finally,the theoretical results are verified through several numerical experiments,and the results calculated on the three grids are compared and analyzed.In Chapter 5,consider the time fractional diffusion equation with Caputo fractional derivative.Observe that the solution of this type of equation is singular at the initial moment,which is difficult to obtain the ideal convergence order on a uniform mesh.Therefore,we establish three non-uniform grids for k=3,4,5 according to the k power formula,denoted as mesh2,meshl,and mesh3.Discrete time fractional derivative using the format L2-1? on a non-uniform grid,where ?=1-?/2;the central difference quotient formula is used to discretize the diffusion term on a uniform grid,and the numerical format of the model equation is established.Through theoretical analysis,we get that the numerical solution format obtained by discretization under different non-uniform grids has different time convergence orders O(N-min{k?,2}),where,N represents the number of time divisions,and the stability of the format is analyzed.Finally,the theoretical analysis results are verified through several numerical examples.By observing the calculation results,we find that the calculation on meshl can get a more accurate solution.For ?>0.5.the calculation on meshl has a second-order convergence rate,which is the best for a graded mesh with adjustable parameters r.In addition,for comparison,we also calculate on the standard graded mesh,which shows that for the case of ?? 0.5,the numerical error of meshl is also better than that of the standard graded mesh.In Chapter 6,for one-dimensional and two-dimensional time fractional diffusion equations with Caputo fractional derivatives,we consider the singularity of the solution at the initial moment.In order to obtain the ideal convergence order,for the Caputo time fractional derivative term,we use the L1-2 format to discretization on the graded meshes and the spatial derivative term is approx-imated by the classical central difference scheme on a uniform mesh.In addition,the local truncation error is estimated for the time-discrete format.Due to the complex positive and negative coefficients in the numerical format,the overal-1 stability analysis of the numerical format is still an unresolved problem.On the other hand,considering the relatively large amount of calculation work for numerical solution,we optimize the direct discretization scheme by means of singular value decomposition and POD technique,and obtain a reducedorder finite-difference extrapolation algorithm.The order reduction algorithm great-ly reduces the number of unknowns in each time layer.The numerical example verifies the convergence of the numerical format,and the time convergence order reaches O(N-min{r?,3-?}).At the same time,the effectiveness of the reducedorder algorithm is verified.The numerical results obtained by the reduced-order finite-difference scheme and the direct-discrete finite-difference scheme are almost the same,and the calculation time of the reduced-order finite-difference scheme is significantly shortened.In Chapter 7,the fractional-order convection-diffusion equation with singularity at the initial moment is studied,and the fast finite difference/RBF meshless method is derived.We first use the classic L1 format to discretize the time derivative term on the graded mesh,and derive the semi-discrete format of the problem.Secondly,the RBF shape function structure is briefly introduced,and then the RBF meshless method is used to discretize the space,and the fully discrete scheme of the fractional convection-diffusion equation is derived.The meshless method does not need to construct the mesh,which is more conducive to dealing with complex areas or complex boundary conditions.However,the meshless method also has the problem of computational efficiency.To solve this problem,we use an proper orthogonal decomposition technique combined with the RBF meshless method to establish a reducedorder meshless extrapolation algorithm for the fractional-order convection-diffusion equation,which has lower dimensionality.Finally,numerical examples of different problem areas and different node distributions are studied,and the finite diference method is used to solve the problem and compared with the RBF meshless method.It is verified that the reduced-order extrapolation meshless method can obtain better accuracy,and effectively save calculation time significantly.In Chapter 8,We summarize the full text and briefly introduce the main research directions in the future.