High order parameter-uniform discretization for singularly perturbed parabolic partial differential equations with time delay

Kumar, Sunil; Kumar, Mukesh

doi:10.1016/j.camwa.2014.09.004

Cited by 58 publications

(44 citation statements)

References 18 publications

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“…To verify the theoretical results established in the previous section, we consider the following two test problems similar to [1,2].…”

Section: Numerical Resultsmentioning

confidence: 92%

“…We first introduce the continuous operator L defined by Lu(x, t) := Lu(x, t)+b(x, t)u(x, t −τ), (x, t) ∈ D, and establish the maximum principle for it. Then, using the maximum principle for the operator L, we derive some a priori bounds for the solution derivatives of problem (1)(2). Similar bounds are derived in [1] using the method of steps and the maximum principle for the operator L.…”

Section: A Priori Boundsmentioning

confidence: 94%

“…So, using In the following theorem, we prove that when σ = 2 � ε α ln N, one iteration is sufficient to provide uniformly convergent approximation of almost second order in space and first order in time. (1)(2) and U [1] be the first iterate of the proposed method. If σ = 2 � ε α ln N, then…”

Section: By Taylor Expansions Lemma 23 and Hmentioning

confidence: 99%

“…To the best of our knowledge no paper is available in literature on domain decomposition for problem (1)(2). Thus, the aim of this article is three-fold:…”

Section: Introductionmentioning

confidence: 99%

“…We derive some a priori bounds for the solution and its derivatives of problem (1-2) using maximum principle in Section 2. We present our method for solving problem (1)(2) in Section 3. The error analysis of the method is established in Section 4.…”

Section: Introductionmentioning

confidence: 99%

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A domain decomposition method for solving singularly perturbed parabolic reaction‐diffusion problems with time delay

Singh

Kumar

2018

Numerical Methods Partial

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We design and analyse a domain decomposition method for solving singularly perturbed parabolic reaction-diffusion problems with time delay. Using the asymptotic behavior of the solution, we decompose the original domain of the problem into three overlapping subdomains, two of which are boundary layer subdomains and one is a regular subdomain. On each subdomain, we discretize the problem by the backward Euler scheme in the time direction and the central difference scheme in the spatial direction. The proposed method is shown to be uniformly convergent, having almost second order in space and first order in time. In addition, we prove that the proposed method converges much faster for small values of perturbation parameter ε. At the end, some numerical results are given in support of theoretical findings.

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“…To verify the theoretical results established in the previous section, we consider the following two test problems similar to [1,2].…”

Section: Numerical Resultsmentioning

confidence: 92%

Section: A Priori Boundsmentioning

confidence: 94%

Section: By Taylor Expansions Lemma 23 and Hmentioning

confidence: 99%

“…To the best of our knowledge no paper is available in literature on domain decomposition for problem (1)(2). Thus, the aim of this article is three-fold:…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A domain decomposition method for solving singularly perturbed parabolic reaction‐diffusion problems with time delay

Singh

Kumar

2018

Numerical Methods Partial

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High‐order schemes and their error analysis for generalized variable coefficients fractional reaction–diffusion equations

Singh

Kumar

Vigo‐Aguiar

2023

Math Methods in App Sciences

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In this manuscript, we develop and analyze two high‐order schemes, CFD and PQS , for generalized variable coefficients fractional reaction–diffusion equations. The generalized fractional derivative is characterized by a weight function and a scale function. We approximate it using generalized Alikhanov formula ( ) of order , where denotes the order of the generalized fractional derivative. Moreover, for spatial discretization, we use a compact operator in CFD scheme and parametric quintic splines in PQS scheme. The stability and convergence analysis of both schemes are demonstrated thoroughly using the discrete energy method in the ‐norm. It is shown that the convergence orders of the CFD and PQS schemes are and , respectively, where and represent the mesh spacing in the time direction and is the mesh spacing in the space direction. In addition, numerical results are obtained for three test problems to validate the theory and demonstrate the efficiency and superiority of the proposed schemes.

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On new approximations of Caputo–Prabhakar fractional derivative and their application to reaction–diffusion problems with variable coefficients

Singh,

Kumar,

Vigo‐Aguiar

2023

Math Methods in App Sciences

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This article is devoted to constructing and analyzing two new approximations (CPL2‐1 and CPL‐2 formulas) for the Caputo–Prabhakar fractional derivative. The error bounds for the CPL2‐1 and CPL‐2 formulas are proved to be of order and , respectively, where is the order of time‐fractional derivative. The newly developed approximations are then used in the numerical treatment of a reaction–diffusion problem with variable coefficients defined in the Caputo–Prabhakar sense. Moreover, the space variable in the developed numerical schemes, CFD1 and CFD2, is discretized using a fourth‐order compact difference operator. Both schemes' stability and convergence analysis are demonstrated thoroughly using the discrete energy method. It is shown that the convergence orders of CFD1 and CFD2 schemes are and , respectively, where and represent the mesh spacing in time and space directions, respectively. In addition, numerical results are obtained for three test problems to confirm the theory and demonstrate the efficiency and superiority of the proposed schemes.

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High order parameter-uniform discretization for singularly perturbed parabolic partial differential equations with time delay

Cited by 58 publications

References 18 publications

A domain decomposition method for solving singularly perturbed parabolic reaction‐diffusion problems with time delay

A domain decomposition method for solving singularly perturbed parabolic reaction‐diffusion problems with time delay

High‐order schemes and their error analysis for generalized variable coefficients fractional reaction–diffusion equations

On new approximations of Caputo–Prabhakar fractional derivative and their application to reaction–diffusion problems with variable coefficients

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