The tempered fractional calculus has been successfully applied for depicting the time evolution of a system describing non-Markovian diffusion particles.The related governing equations are a series of partial differen...The tempered fractional calculus has been successfully applied for depicting the time evolution of a system describing non-Markovian diffusion particles.The related governing equations are a series of partial differential equations with tempered fractional derivatives.Using the polynomial interpolation technique,in this paper,we present three efficient numerical formulas,namely the tempered L1 formula,the tempered L1-2 formula,and the tempered L2-1_(σ)formula,to approximate the Caputo-tempered fractional derivative of orderα∈(0,1).The truncation error of the tempered L1 formula is of order 2-α,and the tempered L1-2 formula and L2-1_(σ)formula are of order 3-α.As an application,we construct implicit schemes and implicit ADI schemes for one-dimensional and two-dimensional time-tempered fractional diffusion equations,respectively.Furthermore,the unconditional stability and convergence of two developed difference schemes with tempered L1 and L2-1_(σ)formulas are proved by the Fourier analysis method.Finally,we provide several numerical examples to demonstrate the correctness and effectiveness of the theoretical analysis.展开更多
Keller-Segel systems are a set of nonlinear partial differential equations used to model chemotaxis in biology.In this paper,we propose two alternating direction implicit(ADI)schemes to solve the 2D Keller-Segel syste...Keller-Segel systems are a set of nonlinear partial differential equations used to model chemotaxis in biology.In this paper,we propose two alternating direction implicit(ADI)schemes to solve the 2D Keller-Segel systems directly with minimal computational cost,while preserving positivity,energy dissipation law and mass conservation.One scheme unconditionally preserves positivity,while the other does so conditionally.Both schemes achieve second-order accuracy in space,with the former being first-order accuracy in time and the latter second-order accuracy in time.Besides,the former scheme preserves the energy dissipation law asymptotically.We validate these results through numerical experiments,and also compare the efficiency of our schemes with the standard five-point scheme,demonstrating that our approaches effectively reduce computational costs.展开更多
A compact alternating direction implicit(ADI) method has been developed for solving multi-dimensional heat equations by introducing the differential operators and the truncation error is O(τ 2 + h 4 ). It is shown by...A compact alternating direction implicit(ADI) method has been developed for solving multi-dimensional heat equations by introducing the differential operators and the truncation error is O(τ 2 + h 4 ). It is shown by the discrete Fourier analysis that this new ADI scheme is unconditionally stable and the truncation error O(τ 3 + h 6 ) is gained with once Richardson's extrapolation. Some numerical examples are presented to demonstrate the efficiency and accuracy of the new scheme.展开更多
This research paper represents a numerical approximation to non-linear two-dimensional reaction diffusion equation from population genetics. Since various initial and boundary value problems exist in two-dimensional r...This research paper represents a numerical approximation to non-linear two-dimensional reaction diffusion equation from population genetics. Since various initial and boundary value problems exist in two-dimensional reaction-diffusion, phenomena are studied numerically by different numerical methods, here we use finite difference schemes to approximate the solution. Accuracy is studied in term of L2, L∞ and relative error norms by random selected grids along time levels for comparison with exact results. The test example demonstrates the accuracy, efficiency and versatility of the proposed schemes. It is shown that the numerical schemes give better solutions. Moreover, the schemes can be easily applied to a wide class of higher dimension nonlinear reaction diffusion equations with a little modification.展开更多
文摘The tempered fractional calculus has been successfully applied for depicting the time evolution of a system describing non-Markovian diffusion particles.The related governing equations are a series of partial differential equations with tempered fractional derivatives.Using the polynomial interpolation technique,in this paper,we present three efficient numerical formulas,namely the tempered L1 formula,the tempered L1-2 formula,and the tempered L2-1_(σ)formula,to approximate the Caputo-tempered fractional derivative of orderα∈(0,1).The truncation error of the tempered L1 formula is of order 2-α,and the tempered L1-2 formula and L2-1_(σ)formula are of order 3-α.As an application,we construct implicit schemes and implicit ADI schemes for one-dimensional and two-dimensional time-tempered fractional diffusion equations,respectively.Furthermore,the unconditional stability and convergence of two developed difference schemes with tempered L1 and L2-1_(σ)formulas are proved by the Fourier analysis method.Finally,we provide several numerical examples to demonstrate the correctness and effectiveness of the theoretical analysis.
基金supported by DOE DE-SC0222766supported by NSF grants DMS-1950868+1 种基金DMS-2153029DMS-2118181。
文摘Keller-Segel systems are a set of nonlinear partial differential equations used to model chemotaxis in biology.In this paper,we propose two alternating direction implicit(ADI)schemes to solve the 2D Keller-Segel systems directly with minimal computational cost,while preserving positivity,energy dissipation law and mass conservation.One scheme unconditionally preserves positivity,while the other does so conditionally.Both schemes achieve second-order accuracy in space,with the former being first-order accuracy in time and the latter second-order accuracy in time.Besides,the former scheme preserves the energy dissipation law asymptotically.We validate these results through numerical experiments,and also compare the efficiency of our schemes with the standard five-point scheme,demonstrating that our approaches effectively reduce computational costs.
文摘A compact alternating direction implicit(ADI) method has been developed for solving multi-dimensional heat equations by introducing the differential operators and the truncation error is O(τ 2 + h 4 ). It is shown by the discrete Fourier analysis that this new ADI scheme is unconditionally stable and the truncation error O(τ 3 + h 6 ) is gained with once Richardson's extrapolation. Some numerical examples are presented to demonstrate the efficiency and accuracy of the new scheme.
文摘This research paper represents a numerical approximation to non-linear two-dimensional reaction diffusion equation from population genetics. Since various initial and boundary value problems exist in two-dimensional reaction-diffusion, phenomena are studied numerically by different numerical methods, here we use finite difference schemes to approximate the solution. Accuracy is studied in term of L2, L∞ and relative error norms by random selected grids along time levels for comparison with exact results. The test example demonstrates the accuracy, efficiency and versatility of the proposed schemes. It is shown that the numerical schemes give better solutions. Moreover, the schemes can be easily applied to a wide class of higher dimension nonlinear reaction diffusion equations with a little modification.
