Numerical scheme with convergence for a generalized time‐fractional Telegraph‐type equation

Kumar, Kamlesh; Pandey, Rajesh K.; Sharma, Shiva; Xu, Yufeng

doi:10.1002/num.22344

Cited by 14 publications

(3 citation statements)

References 28 publications

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“…Fractional differential equations originated in 1695, see [1,2]. As is known, they can provide excellent descriptive models to resolve various problems in reality, and they are applied in various fields, such as control engineering [3], viscoelastic materials [4], fluid mechanics [5], electrochemistry [6], the analysis of epidemic [7] and complex networks [8], statistical mechanics [9], numerical schemes [10], etc. The relevant problems for the diffusion equations have been studied by many scholars, see [11,12].…”

Section: Introductionmentioning

confidence: 99%

Mild Solutions of Fractional Integrodifferential Diffusion Equations with Nonlocal Initial Conditions via the Resolvent Family

Mu,

Yuan,

Zhou

2023

Fractal Fract

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Fractional integrodifferential diffusion equations play a significant role in describing anomalous diffusion phenomena. In this paper, we study the existence and uniqueness of mild solutions to these equations. Firstly, we construct an appropriate resolvent family, through which the related equicontinuity, strong continuity, and compactness properties are studied using the convolution theorem of Laplace transform, the probability density function, the Cauchy integral formula, and the Fubini theorem. Then, we construct a reasonable mild solution for the considered equations. Finally, we obtain some sufficient conditions for the existence and uniqueness of mild solutions to the considered equations by some fixed point theorems.

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Section: Introductionmentioning

confidence: 99%

Mild Solutions of Fractional Integrodifferential Diffusion Equations with Nonlocal Initial Conditions via the Resolvent Family

Mu,

Yuan,

Zhou

2023

Fractal Fract

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“…The numerical solutions to these problems were obtained using the finite difference method. Further, the authors of [17,18], studied the time-fractional telegraph equations and fractional advection-diffusion equations in terms of GFDs and developed the higher order schemes to solve such equations. Kumar et al [19] presented two numerical schemes to approximate the GFDs and obtained the convergence orders as (2 − α) and (3 − α), respectively.…”

Section: Introductionmentioning

confidence: 99%

Finite Difference–Collocation Method for the Generalized Fractional Diffusion Equation

et al. 2022

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In this paper, an approximate method combining the finite difference and collocation methods is studied to solve the generalized fractional diffusion equation (GFDE). The convergence and stability analysis of the presented method are also established in detail. To ensure the effectiveness and the accuracy of the proposed method, test examples with different scale and weight functions are considered, and the obtained numerical results are compared with the existing methods in the literature. It is observed that the proposed approach works very well with the generalized fractional derivatives (GFDs), as the presence of scale and weight functions in a generalized fractional derivative (GFD) cause difficulty for its discretization and further analysis.

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“…Kumar et al [34] presented a numerical scheme for the generalized fractional telegraph equation in time.…”

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confidence: 99%

High-Order Approximation to Generalized Caputo Derivatives and Generalized Fractional Advection–Diffusion Equations

2023

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In this article, a high-order time-stepping scheme based on the cubic interpolation formula is considered to approximate the generalized Caputo fractional derivative (GCFD). Convergence order for this scheme is (4−α), where α(0<α<1) is the order of the GCFD. The local truncation error is also provided. Then, we adopt the developed scheme to establish a difference scheme for the solution of the generalized fractional advection–diffusion equation with Dirichlet boundary conditions. Furthermore, we discuss the stability and convergence of the difference scheme. Numerical examples are presented to examine the theoretical claims. The convergence order of the difference scheme is analyzed numerically, which is (4−α) in time and second-order in space.

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Numerical scheme with convergence for a generalized time‐fractional Telegraph‐type equation

Cited by 14 publications

References 28 publications

Mild Solutions of Fractional Integrodifferential Diffusion Equations with Nonlocal Initial Conditions via the Resolvent Family

Mild Solutions of Fractional Integrodifferential Diffusion Equations with Nonlocal Initial Conditions via the Resolvent Family

Finite Difference–Collocation Method for the Generalized Fractional Diffusion Equation

High-Order Approximation to Generalized Caputo Derivatives and Generalized Fractional Advection–Diffusion Equations

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