Difference schemes for the numerical solution of the heat conduction equation with aftereffect

Пименов, В. Г.; Lozhnikov, A. B.

doi:10.1134/s0081543811090100

Cited by 20 publications

(29 citation statements)

References 6 publications

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“…By the method of embedding a scheme with a weight into the general difference scheme with aftereffect [10] we prove the following theorem. …”

Section: Theoremmentioning

confidence: 99%

“…Let us consider (7) from the point of view of operator-difference equations and apply methods of the stability verification of a two-level difference scheme [9] and the separation of finite-dimensional and infinite-dimensional components [8,10].…”

Section: Theoremmentioning

confidence: 99%

“…Since B is invertible, we can write (7) in the form y j+1 = Sy j + ΔB −1 F j , where S = (E − ΔB −1 A) is the so-called [8,10] operator of moving by a step.…”

Section: Introduce a Vector Ymentioning

confidence: 99%

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A difference scheme for the numerical solution of an advection equation with aftereffect

Solodushkin

2013

Russ Math.

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Abstract-We propose a family of grid methods for the numerical solution of an advection equation with a time delay in a general form. The methods are based on the idea of separating the current state and the prehistory function. We prove the convergence of the second-order method coordinatewise and do that of the first-order with respect to time. The proof is based on techniques applied for proving analogous theorems for functional differential equations and on the general theory of difference schemes. We illustrate the obtained results with a test example.

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“…By the method of embedding a scheme with a weight into the general difference scheme with aftereffect [10] we prove the following theorem. …”

Section: Theoremmentioning

confidence: 99%

Section: Theoremmentioning

confidence: 99%

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A difference scheme for the numerical solution of an advection equation with aftereffect

Solodushkin

2013

Russ Math.

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“…P r o o f. Since the piecewise linear interpolation operator with extrapolation by continuation has second order [8], therefore there exists a constant C, such that for all t…”

Section: Lemma 2 Assume That the Conditions Of Lemma 1 Are Satisfiedmentioning

confidence: 99%

“…Numerical methods for solving such equations were considered in many papers, for example [3][4][5][6][7]. In paper [8], a technique of study on stability and convergence of numerical algorithms using the general theory of differential schemes was constructed for the heat conduction equation with delay [9] and the theory of numerical methods of the solution of the functional and differential equations were discussed in [10,11]. After that, this technique was applied to research of numerical methods of the solution of the equations of hyperbolic type with delay [12], various equations of parabolic type [13,14] and other types of the equations in partial derivatives with effect of heredity [15].…”

Section: Introductionmentioning

confidence: 99%

A Fractional Analog of Crank–nicholson Method for the Two Sided Space Fractional Partial Equation With Functional Delay

Пименов

Hendy

2016

UMJ

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Convergence and stability estimates in difference setting for time‐fractional parabolic equations with functional delay

Hendy

Пименов

Macías‐Díaz

2019

Numerical Methods Partial

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A class of one‐dimensional time‐fractional parabolic differential equations with delay effects of functional type in the time component is numerically investigated in this work. To that end, a compact difference scheme is constructed for the numerical solution of those equations based on the idea of separating the current state and the prehistory function. In these terms, the prehistory function is approximated by means of an appropriate interpolation–extrapolation operator. A discrete form of the fractional Gronwall inequality is employed to provide an optimal error estimate. The existence and uniqueness of the numerical solutions, the order of approximation error for the constructed scheme, the stability and the order of convergence are mathematically investigated in this work.

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Difference schemes for the numerical solution of the heat conduction equation with aftereffect

Cited by 20 publications

References 6 publications

A difference scheme for the numerical solution of an advection equation with aftereffect

A difference scheme for the numerical solution of an advection equation with aftereffect

A Fractional Analog of Crank–nicholson Method for the Two Sided Space Fractional Partial Equation With Functional Delay

Convergence and stability estimates in difference setting for time‐fractional parabolic equations with functional delay

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