Linear fractional map type (LFMT) nonlinear QCA (NLQCA), one of the simplest reversible NLQCA is studied analytically as well as numerically. Linear advection equation or Time Dependent Schrödinger Equation (...Linear fractional map type (LFMT) nonlinear QCA (NLQCA), one of the simplest reversible NLQCA is studied analytically as well as numerically. Linear advection equation or Time Dependent Schrödinger Equation (TDSE) is obtained from the continuum limit of linear QCA. Similarly it is found that some nonlinear advection-diffusion equations including inviscid Burgers equation and porous-medium equation are obtained from LFMT NLQCA.展开更多
This paper is concerned with the following variable-order fractional Laplacian equations , where N ≥ 1 and N > 2s(x,y) for (x,y) ∈ Ω × Ω, Ω is a bounded domain in R<sup>N</sup>, s(⋅)...This paper is concerned with the following variable-order fractional Laplacian equations , where N ≥ 1 and N > 2s(x,y) for (x,y) ∈ Ω × Ω, Ω is a bounded domain in R<sup>N</sup>, s(⋅) ∈ C (R<sup>N</sup> × R<sup>N</sup>, (0,1)), (-Δ)<sup>s(⋅)</sup> is the variable-order fractional Laplacian operator, λ, μ > 0 are two parameters, V: Ω → [0, ∞) is a continuous function, f ∈ C(Ω × R) and q ∈ C(Ω). Under some suitable conditions on f, we obtain two solutions for this problem by employing the mountain pass theorem and Ekeland’s variational principle. Our result generalizes the related ones in the literature.展开更多
The higher-order numerical scheme of nonlinear advection-diffusion equations is studied in this article, where the space fractional derivatives are evaluated by using weighted and shifted Grünwald difference oper...The higher-order numerical scheme of nonlinear advection-diffusion equations is studied in this article, where the space fractional derivatives are evaluated by using weighted and shifted Grünwald difference operators and combining the compact technique, in the time direction is discretized by the Crank-Nicolson method. Through the energy method, the stability and convergence of the numerical scheme in the sense of L<sub>2</sub>-norm are proved, and the convergence order is . Some examples are given to show that our numerical scheme is effective.展开更多
In this article, a finite volume element algorithm is presented and discussed for the numerical solutions of a time-fractional nonlinear fourth-order diffusion equation with time delay. By choosing the second-order sp...In this article, a finite volume element algorithm is presented and discussed for the numerical solutions of a time-fractional nonlinear fourth-order diffusion equation with time delay. By choosing the second-order spatial derivative of the original unknown as an additional variable, the fourth-order problem is transformed into a second-order system. Then the fully discrete finite volume element scheme is formulated by using L1approximation for temporal Caputo derivative and finite volume element method in spatial direction. The unique solvability and stable result of the proposed scheme are proved. A priori estimate of L2-norm with optimal order of convergence O(h2+τ2−α)where τand hare time step length and space mesh parameter, respectively, is obtained. The efficiency of the scheme is supported by some numerical experiments.展开更多
In this work,a novel time-stepping L1 formula is developed for a hidden-memory variable-order Caputo’s fractional derivative with an initial singularity.This formula can obtain second-order accuracy and an error esti...In this work,a novel time-stepping L1 formula is developed for a hidden-memory variable-order Caputo’s fractional derivative with an initial singularity.This formula can obtain second-order accuracy and an error estimate is analyzed strictly.As an application,a fully discrete difference scheme is established for the initial-boundary value problem of a hidden-memory variable-order time fractional diffusion model.Numerical experiments are provided to support our theoretical results.展开更多
In this study,we investigate a mathematical model that describes the growth dynamics of glial cells in glioma,formulated as a nonlinear partial differential equation with a treatment-dependent source term.To approxima...In this study,we investigate a mathematical model that describes the growth dynamics of glial cells in glioma,formulated as a nonlinear partial differential equation with a treatment-dependent source term.To approximate the solution of this model,we employ three semi-analytical techniques:the Homotopy Analysis Method(HAM),the Homotopy Perturbation Method(HPM),and the Reduced Differential Transform Method(RDTM).A comparative analysis shows that while all three methods produce accurate results,RDTM exhibits rapid stabilization across various time points,outperforming HAM and HPM in terms of convergence speed and computational efficiency.To incorporate memory effects commonly observed in biological systems,we extend the model to a fractional-order framework.Within this extension,we apply HPM,the Fractional Reduced Differential Transform Method(FRDTM),and RDTM to construct higher-order approximations and examine their convergence behavior.We also conduct detailed convergence and error analysis for the resulting series of solutions,providing theoretical validation of their accuracy and reliability.The simulation results reveal a steady decline in glial cell concentration over time,eventually approaching negligible levels,indicating effective suppression of glioma growth under the modeled treatment.Notably,smaller values of the fractional-order parameter accelerate this decline,highlighting the significant influence of fractional dynamics on treatment outcomes.Finally,we establish the existence,uniqueness,and stability of the solution using the sectorial operator framework and the Mittag-Leffler function representation,reinforcing the mathematical soundness of the proposed model.These findings underscore the potential of fractional modeling and semi-analytical methods in capturing the complex behavior of glioma progression and enhancing therapeutic strategies.展开更多
文摘Linear fractional map type (LFMT) nonlinear QCA (NLQCA), one of the simplest reversible NLQCA is studied analytically as well as numerically. Linear advection equation or Time Dependent Schrödinger Equation (TDSE) is obtained from the continuum limit of linear QCA. Similarly it is found that some nonlinear advection-diffusion equations including inviscid Burgers equation and porous-medium equation are obtained from LFMT NLQCA.
文摘This paper is concerned with the following variable-order fractional Laplacian equations , where N ≥ 1 and N > 2s(x,y) for (x,y) ∈ Ω × Ω, Ω is a bounded domain in R<sup>N</sup>, s(⋅) ∈ C (R<sup>N</sup> × R<sup>N</sup>, (0,1)), (-Δ)<sup>s(⋅)</sup> is the variable-order fractional Laplacian operator, λ, μ > 0 are two parameters, V: Ω → [0, ∞) is a continuous function, f ∈ C(Ω × R) and q ∈ C(Ω). Under some suitable conditions on f, we obtain two solutions for this problem by employing the mountain pass theorem and Ekeland’s variational principle. Our result generalizes the related ones in the literature.
文摘The higher-order numerical scheme of nonlinear advection-diffusion equations is studied in this article, where the space fractional derivatives are evaluated by using weighted and shifted Grünwald difference operators and combining the compact technique, in the time direction is discretized by the Crank-Nicolson method. Through the energy method, the stability and convergence of the numerical scheme in the sense of L<sub>2</sub>-norm are proved, and the convergence order is . Some examples are given to show that our numerical scheme is effective.
文摘In this article, a finite volume element algorithm is presented and discussed for the numerical solutions of a time-fractional nonlinear fourth-order diffusion equation with time delay. By choosing the second-order spatial derivative of the original unknown as an additional variable, the fourth-order problem is transformed into a second-order system. Then the fully discrete finite volume element scheme is formulated by using L1approximation for temporal Caputo derivative and finite volume element method in spatial direction. The unique solvability and stable result of the proposed scheme are proved. A priori estimate of L2-norm with optimal order of convergence O(h2+τ2−α)where τand hare time step length and space mesh parameter, respectively, is obtained. The efficiency of the scheme is supported by some numerical experiments.
基金supported by the National Natural Science Foundation of China(No.12201076)the China Postdoctoral Science Foundation(No.2023M732180)。
文摘In this work,a novel time-stepping L1 formula is developed for a hidden-memory variable-order Caputo’s fractional derivative with an initial singularity.This formula can obtain second-order accuracy and an error estimate is analyzed strictly.As an application,a fully discrete difference scheme is established for the initial-boundary value problem of a hidden-memory variable-order time fractional diffusion model.Numerical experiments are provided to support our theoretical results.
文摘In this study,we investigate a mathematical model that describes the growth dynamics of glial cells in glioma,formulated as a nonlinear partial differential equation with a treatment-dependent source term.To approximate the solution of this model,we employ three semi-analytical techniques:the Homotopy Analysis Method(HAM),the Homotopy Perturbation Method(HPM),and the Reduced Differential Transform Method(RDTM).A comparative analysis shows that while all three methods produce accurate results,RDTM exhibits rapid stabilization across various time points,outperforming HAM and HPM in terms of convergence speed and computational efficiency.To incorporate memory effects commonly observed in biological systems,we extend the model to a fractional-order framework.Within this extension,we apply HPM,the Fractional Reduced Differential Transform Method(FRDTM),and RDTM to construct higher-order approximations and examine their convergence behavior.We also conduct detailed convergence and error analysis for the resulting series of solutions,providing theoretical validation of their accuracy and reliability.The simulation results reveal a steady decline in glial cell concentration over time,eventually approaching negligible levels,indicating effective suppression of glioma growth under the modeled treatment.Notably,smaller values of the fractional-order parameter accelerate this decline,highlighting the significant influence of fractional dynamics on treatment outcomes.Finally,we establish the existence,uniqueness,and stability of the solution using the sectorial operator framework and the Mittag-Leffler function representation,reinforcing the mathematical soundness of the proposed model.These findings underscore the potential of fractional modeling and semi-analytical methods in capturing the complex behavior of glioma progression and enhancing therapeutic strategies.
