The OSC solver for the fourth-order sub-diffusion equation with weakly singular solutions

Yang, Xuehua; Zhang, Haixiang; Tang, Jie

doi:10.1016/j.camwa.2020.11.015

Cited by 48 publications

(10 citation statements)

References 23 publications

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“…To show this, it suffices to introduce another test function, say ψ, which is continuously differentiable on [0, T] with values ψ(0) = 1 and ψ(T) = 0. With these claims in place, we proceed according to the variational Formulation (11)…”

Section: Compactness Methods and Passage To The Limitmentioning

confidence: 99%

“…The above uniform estimate of the solution of the problem is addressed first here, and all constants independent of m will be denoted by C. With this, we proceed by asserting for simplicity and notational sake that if u m is replaced by u, we can show that u m is uniformly bounded in the space. The above claim is shown by setting v = u in Equation (11) to have 1 2…”

Section: Uniform Estimates Of the Solution Of The Problemmentioning

confidence: 99%

“…Many powerful mathematical techniques have been used in recent years by researchers to solve differential equations. As a result of continuous research efforts, a great number of efficient methods for solving differential equations have been developed with various forms of discretizations [7][8][9][10][11]. Another efficient technique is the Adomian decomposition method (ADM) found in [3,12], which yields analytic solutions in the form of rapidly convergent infinite power series with easily computable terms.…”

Section: Introductionmentioning

confidence: 99%

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An Analysis of the Nonstandard Finite Difference and Galerkin Methods Applied to the Huxley Equation

Chin,

Moutsinga,

Adem

2024

Mathematics

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The Huxley equation, which is a nonlinear partial differential equation, is used to describe the ionic mechanisms underlying the initiation and propagation of action potentials in the squid giant axon. This equation, just like many other nonlinear equations, is often very difficult to analyze because of the presence of the nonlinearity term, which is always very difficult to approximate. This paper aims to design a reliable scheme that consists of a combination of the nonstandard finite difference in time method, the Galerkin method and the compactness methods in space variables. This method is used to show that the solution of the problem exists uniquely. The a priori estimate from the existence process is applied to the scheme to show that the numerical solution from the scheme converges optimally in the L2 as well as the H1 norms. We proceed to show that the scheme preserves the decaying properties of the exact solution. Numerical experiments are introduced with a chosen example to validate the proposed theory.

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Section: Compactness Methods and Passage To The Limitmentioning

confidence: 99%

Section: Uniform Estimates Of the Solution Of The Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An Analysis of the Nonstandard Finite Difference and Galerkin Methods Applied to the Huxley Equation

Chin,

Moutsinga,

Adem

2024

Mathematics

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“…In addition, Yang et al [36] verified uniform

{l}^1

long time convergence by combining the first‐order CQ rule and the L1 formula. With the development of numerical research on this problem, new studies have emerged recently, such as the Sinc collocation method [21, 22, 37], meshless technique [1], orthogonal spline collocation method [35] and time two‐grid method [3, 34], see the references therein for other works.…”

Section: Introductionmentioning

confidence: 99%

Efficient and accurate temporal second‐order numerical methods for multidimensional multi‐term integrodifferential equations with the Abel kernels

Zhao,

Chen,

2023

Numerical Methods Partial

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This work develops two temporal second‐order alternating direction implicit (ADI) numerical schemes for solving multidimensional parabolic‐type integrodifferential equations with multi‐term weakly singular Abel kernels. For the two‐dimensional (2D) case, applying the Crank–Nicolson method and product integration rule to discretizations of temporal derivative and integral terms, respectively, and the spatial discretization is proposed using a compact difference formulation combined with the ADI algorithm; for the three‐dimensional case, the method of temporal discretization is the same as the 2D case, and then we employ the standard finite difference in space to construct a fully discrete ADI finite difference scheme. The ADI technique is used to reduce the calculation cost of the high‐dimensional problem. Besides, the stability and convergence of two ADI schemes are rigorously proved by the energy argument, in which the first scheme converges to the order , where , , and denote the time‐space step sizes, respectively, and the second scheme converges to the space‐time second‐order accuracy. Finally, the numerical results verify the correctness of the theoretical analysis and show that the method of this article is competitive with the existing research work.

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“…Most of the numerical techniques opted in the articles are based upon L1 approximation of Caputo time derivative which gives first-order convergence for fractional order 0 < α ≤ 1, where α is the fractional order (present in time-fractional derivative). Recently, articles based on orthogonal spline collocation method are published [39] which focus on weakly singular solutions and constructing a high-order numerical scheme for fourth-order subdiffusion equations.…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2021

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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.

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The OSC solver for the fourth-order sub-diffusion equation with weakly singular solutions

Cited by 48 publications

References 23 publications

An Analysis of the Nonstandard Finite Difference and Galerkin Methods Applied to the Huxley Equation

An Analysis of the Nonstandard Finite Difference and Galerkin Methods Applied to the Huxley Equation

Efficient and accurate temporal second‐order numerical methods for multidimensional multi‐term integrodifferential equations with the Abel kernels

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

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