Alikhanov Legendre–Galerkin spectral method for the coupled nonlinear time-space fractional Ginzburg–Landau complex system

A finite difference/Galerkin spectral discretization for the temporal and spatial fractional coupled Ginzburg-Landau system is proposed and analyzed. The Alikhanov L2-1 sigma difference formula is utilized to discretize the time Caputo fractional derivative, while the Legendre-Galerkin spectral approximation is used to approximate the Riesz spatial fractional operator. The scheme is shown efficiently applicable with spectral accuracy in space and second-order in time. A discrete form of the fractional Gronwall inequality is applied to establish the error estimates of the approximate solution based on the discrete energy estimates technique. The key aspects of the implementation of the numerical continuation are complemented with some numerical experiments to confirm the theoretical claims

Optimal error analysis of a non-uniform IMEX-L1 finite element method for time fractional PDEs and PIDEs

Stability and optimal convergence analysis of a non-uniform implicit-explicit L1 finite element method (IMEX-L1-FEM) is studied for a class of time-fractional linear partial differential/integro-differential equations with non-self-adjoint elliptic part having (space-time) variable coefficients. The proposed scheme is based on a combination of an IMEX-L1 method on graded mesh in the temporal direction and a finite element method in the spatial direction. With the help of a discrete fractional Gr\"{o}nwall inequality, optimal error estimates in $L^2$- and $H^1$-norms are derived for the problem with initial data $u_0 \in H_0^1(\Omega)\cap H^2(\Omega)$. Under higher regularity condition $u_0 \in \dot{H}^3(\Omega)$, a super convergence result is established and as a consequence, $L^\infty$ error estimate is obtained for 2D problems. Numerical experiments are presented to validate our theoretical findings.Comment: 33 page

A discrete Grönwall inequality and energy estimates in the analysis of a discrete model for a nonlinear time-fractional heat equation

In the present work, we investigate the efficiency of a numerical scheme to solve a nonlinear time-fractional heat equation with sufficiently smooth solutions, which was previously reported in the literature [Fract. Calc. Appl. Anal. 16: 892-910 (2013)]. In that article, the authors established the stability and consistency of the discrete model using arguments from Fourier analysis. As opposed to that work, in the present work, we use the method of energy inequalities to show that the scheme is stable and converges to the exact solution with order O(τ2-α + h4), in the case that 0 < α < 1 satisfies 3α ≥ 3/2, which means that 0.369 α ≤ 1. The novelty of the present work lies in the derivation of suitable energy estimates, and a discrete fractional Grönwall inequality, which is consistent with the discrete approximation of the Caputo fractional derivative of order 0 < α < 1 used for that scheme at tk+1/2. © 2020 by the authors.The first author wishes to acknowledge the support of RFBR Grant 19-01-00019. Meanwhile, the second author would like to acknowledge the financial support of the National Council for Science and Technology of Mexico (CONACYT). The second author acknowledges financial support from CONACYT through grant A1-S-45928. Acknowledgments: The authors wish to thank the guest editors for their kind invitation to submit a paper to the special issue of Mathematics MDPI on "Computational Mathematics and Neural Systems". They also wish to thank the anonymous reviewers for their comments and criticisms. All of their comments were taken into account in the revised version of the paper, resulting in a substantial improvement with respect to the original submission

Numerical Simulation for a Multidimensional Fourth-Order Nonlinear Fractional Subdiffusion Model with Time Delay

The purpose of this paper is to develop a numerical scheme for the two-dimensional fourth-order fractional subdiffusion equation with variable coefficients and delay. Using the L2 − 1σ approximation of the time Caputo derivative, a finite difference method with second-order accuracy in the temporal direction is achieved. The novelty of this paper is to introduce a numerical scheme for the problem under consideration with variable coefficients, nonlinear source term, and delay time constant. The numerical results show that the global convergence orders for spatial and time dimensions are approximately fourth order in space and second-order in time. © 2021 by the authors. Licensee MDPI, Basel, Switzerland.Acknowledgments: M.A.Z. wishes to acknowledge the support of Nazarbayev University Program 091019CRP2120 and the partial support of the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant “Dynamical Analysis and Synchronization of Complex Neural Networks with Its Applications”). M.A.Z. wishes also to acknowledge the financial support of the National Research Centre of Egypt (NRC)

An Efficient Hybrid Numerical Scheme for Nonlinear Multiterm Caputo Time and Riesz Space Fractional-Order Diffusion Equations with Delay

In this paper, we construct and analyze a linearized finite difference/Galerkin-Legendre spectral scheme for the nonlinear multiterm Caputo time fractional-order reaction-diffusion equation with time delay and Riesz space fractional derivatives. The temporal fractional orders in the considered model are taken as 0<β0<β1<β2<⋯<βm<1. The problem is first approximated by the L1 difference method on the temporal direction, and then, the Galerkin-Legendre spectral method is applied on the spatial discretization. Armed by an appropriate form of discrete fractional Grönwall inequalities, the stability and convergence of the fully discrete scheme are investigated by discrete energy estimates. We show that the proposed method is stable and has a convergent order of 2-βm in time and an exponential rate of convergence in space. We finally provide some numerical experiments to show the efficacy of the theoretical results. © 2021 A. K. Omran et al.A. K. Omran is funded by a scholarship under the joint executive program between the Arab Republic of Egypt and Russian Federation. M. A. Zaky wishes to acknowledge the support of the Nazarbayev University Program (091019CRP2120). M. A. Zaky wishes also to acknowledge the partial support of the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant “Dynamical Analysis and Synchronization of Complex Neural Networks with Its Applications”)

Numerical study of ergodicity for the overdamped Generalized Langevin Equation with fractional noise

The Generalized Langevin Equation, in history, arises as a natural fix for the rather traditional Langevin equation when the random force is no longer memoryless. It has been proved that with fractional Gaussian noise (fGn) mostly considered by biologists, the overdamped Generalized Langevin equation satisfying fluctuation-dissipation theorem can be written as a fractional stochastic differential equation (FSDE). While the ergodicity is clear for linear forces, it remains less transparent for nonlinear forces. In this work, we present both a direct and a fast algorithm respectively to this FSDE model. The strong orders of convergence are proved for both schemes, where the role of the memory effects can be clearly observed. We verify the convergence theorems using linear forces, and then present the ergodicity study of the double well potentials in both 1D and 2D setups

Error Estimates of a Continuous Galerkin Time Stepping Method for Subdiffusion Problem

From Springer Nature via Jisc Publications RouterHistory: received 2021-02-12, rev-recd 2021-05-23, accepted 2021-07-06, registration 2021-07-11, pub-electronic 2021-07-29, online 2021-07-29, pub-print 2021-09Publication status: PublishedAbstract: A continuous Galerkin time stepping method is introduced and analyzed for subdiffusion problem in an abstract setting. The approximate solution will be sought as a continuous piecewise linear function in time t and the test space is based on the discontinuous piecewise constant functions. We prove that the proposed time stepping method has the convergence order O(τ1+α), α∈(0, 1) for general sectorial elliptic operators for nonsmooth data by using the Laplace transform method, where τ is the time step size. This convergence order is higher than the convergence orders of the popular convolution quadrature methods (e.g., Lubich’s convolution methods) and L-type methods (e.g., L1 method), which have only O(τ) convergence for the nonsmooth data. Numerical examples are given to verify the robustness of the time discretization schemes with respect to data regularity

Controlo ótimo fracionário e aplicações biológicas

In this PhD thesis, we derive a Pontryagin Maximum Principle (PMP) for fractional optimal control problems and analyze a fractional mathematical model of COVID– 19 transmission dynamics. Fractional optimal control problems consist on optimizing a performance index functional subject to a fractional control system. One of the most important results in optimal control is the Pontryagin Maximum Principle, which gives a necessary optimality condition that every solution to the optimization problem must verify. First, we study properties of optimality for a dynamical system described by distributed-order non-local derivatives associated to a Lagrangian cost functional. We start by proving continuity and differentiability of solutions due to control perturbations. For smooth and unconstrained data, we obtain a weak version of Pontryagin's Maximum principle and a sufficient optimality condition under appropriate convexity. However, for controls taking values on a closed set, we use needle like variations to prove a strong version of Pontryagin's maximum principle. In the second part of the thesis, optimal control problems for fractional operators involving general analytic kernels are studied. We prove an integration by parts formula and a Gronwall inequality for fractional derivatives with a general analytic kernel. Based on these results, we show continuity and differentiability of solutions due to control perturbations leading to a weak version of the maximum principle. In addition, a wide class of combined fractional operators with general analytic kernels is considered. For this later problem, the control set is a closed convex subset of L2. Thus, using techniques from variational analysis, optimality conditions of Pontryagin type are obtained. Lastly, a fractional model for the COVID--19 pandemic, describing the realities of Portugal, Spain and Galicia, is studied. We show that the model is mathematically and biologically well posed. Then, we obtain a result on the global stability of the disease free equilibrium point. At the end we perform numerical simulations in order to illustrate the stability and convergence to the equilibrium point. For the data of Wuhan, Galicia, Spain, and Portugal, the order of the Caputo fractional derivative in consideration takes different values, characteristic of each region, which are not close to one, showing the relevance of the considered fractional models. 2020 Mathematics Subject Classification: 26A33, 49K15, 34A08, 34D23, 92D30.Nesta tese, derivamos o Princípio do Máximo de Pontryagin (PMP) para problemas de controlo ótimo fracionário e analisamos um modelo matemático fracionário para a dinâmica de transmissão da COVID-19. Os problemas de controlo ótimo fracionário consistem em otimizar uma funcional de índice de desempenho sujeita a um sistema de controlo fracionário. Um dos resultados mais importantes no controlo ótimo é o Princípio do Máximo de Pontryagin, que fornece uma condição de otimalidade necessária que toda a solução para o problema de otimização deve verificar. Primeiramente, estudamos propriedades de otimalidade para sistemas dinâmicos descritos por derivadas não-locais de ordem distribuída associadas a uma funcional de custo Lagrangiana. Começamos demonstrando a continuidade e a diferenciabilidade das soluções usando perturbações do controlo. Para dados suaves e sem restrições, obtemos uma versão fraca do princípio do Máximo de Pontryagin e uma condição de otimalidade suficiente sob convexidade apropriada. No entanto, para controlos que tomam valores num conjunto fechado, usamos variações do tipo agulha para provar uma versão forte do princípio do máximo de Pontryagin. Na segunda parte da tese, estudamos problemas de controlo ótimo para operadores fracionários envolvendo um núcleo analítico geral. Demonstramos uma fórmula de integração por partes e uma desigualdade Gronwall para derivadas fracionárias com um núcleo analítico geral. Com base nesses resultados, mostramos a continuidade e a diferenciabilidade das soluções por perturbações do controlo, levando a uma formulação de uma versão fraca do princípio do máximo de Pontryagin. Além disso, consideramos uma classe ampla de operadores fracionários combinados com núcleo analítico geral. Para este último problema, o conjunto de controlos é um subconjunto convexo fechado de L2. Assim, usando técnicas da análise variacional, obtemos condições de otimalidade do tipo de Pontryagin. Finalmente, estudamos um modelo fracionário da pandemia de COVID-19, descrevendo as realidades de Portugal, Espanha e Galiza. Mostramos que o modelo proposto é matematicamente e biologicamente bem colocado. Então, obtemos um resultado sobre a estabilidade global do ponto de equilíbrio livre de doença. No final, realizamos simulações numéricas para ilustrar a estabilidade e convergência do ponto de equilíbrio. Para os dados de Wuhan, Galiza, Espanha e Portugal, a ordem da derivada fracionária de Caputo em consideração toma valores diferentes característicos de cada região, e não próximos de um, mostrando a relevância de se considerarem modelos fracionários.Programa Doutoral em Matemática Aplicad