Design and stability analysis of an implicit non-standard finite difference scheme for fractional neutron point kinetic equation

Roul, Pradip; Goura, V.M.K. Prasad; Madduri, Harshita; Obaidurrahman, K.

doi:10.1016/j.apnum.2019.05.029

Cited by 29 publications

(11 citation statements)

References 27 publications

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“…For DE models, the use of non‐standard finite difference (NSFD) methods 7 has experienced an increase in the last years, 8‐11 in part for the possibility of designing methods with appropriate dynamic consistency properties 12 . A field of applications where these characteristics of dynamic consistency are particularly desirable is in epidemiology modeling, where NSFD methods have been applied both to DE and DDE models 13‐18 …”

Section: Introductionmentioning

confidence: 99%

Nonstandard finite difference schemes for general linear delay differential systems

Castro

Sirvent

Rodríguez

2020

Math Methods in App Sciences

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This paper deals with the construction of non‐standard finite difference methods for coupled linear delay differential systems in the general case of non‐commuting matrix coefficients. Based on an expression for the exact solution of the continuous initial value vector delay problem, a family of non‐standard numerical methods of increasing orders is defined. Numerical examples show that the new proposed numerical methods preserve the stability properties of the exact solutions. This work extends previous results for delay systems with commuting matrix coefficients, allowing the use of the new numerical non‐standard methods in more general problems.

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Section: Introductionmentioning

confidence: 99%

Nonstandard finite difference schemes for general linear delay differential systems

Castro

Sirvent

Rodríguez

2020

Math Methods in App Sciences

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“…This is due to their wide‐range of applications in various fields of applied science and engineering. To mention few, the areas of applications of FDEs include signal processing, viscoelasticity, electrical network, continuum mechanics, material science and nuclear science, see in previous works 1–12 . Moreover, the FDEs were employed to model financial problems 13–15 …”

Section: Introductionmentioning

confidence: 99%

“…To mention few, the areas of applications of FDEs include signal processing, viscoelasticity, electrical network, continuum mechanics, material science and nuclear science, see in previous works. [1][2][3][4][5][6][7][8][9][10][11][12] Moreover, the FDEs were employed to model financial problems. [13][14][15] It should be noted that the classical Black-Scholes (BS) model 16,17 was derived by imposing quite restrictive assumptions, for example, stock pays no dividend, interest rates are constant, transaction costs are eliminated and the values of short-term rates are available.…”

Section: Introductionmentioning

confidence: 99%

Design and analysis of a high order computational technique for time‐fractional Black–Scholes model describing option pricing

Roul

2022

Math Methods in App Sciences

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This work deals with the construction and analysis of a high‐order computational scheme for a time‐fractional Black‐Scholes model that governs the European option pricing. The time‐fractional derivative is considered in the sense of Caputo and the L1 − 2 formula is employed to approximate the Caputo temporal‐fractional derivative of order α, where α ∈ (0, 1). A compact difference scheme is designed for discretization of space variable. The convergence of the method is discussed in detail. It is shown that the proposed method has fourth order accuracy in space and false(3−αfalse)−th order accuracy in time. One numerical example with the known exact solution is considered to demonstrate the applicability and accuracy of present numerical scheme. Moreover, the suggested numerical scheme is employed for pricing three European option problems, namely European call option, European double barrier knock‐out option and European put option. The effect of fractional order derivative on option price profile is investigated. Furthermore, the effects of three relevant parameters, namely volatility, strike price and interest rate on the price of European double barrier knock‐out option are investigated. The computational time for the proposed method is provided.

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“…The study of fractional differential equations has gained much importance in recent years due to their applications in numerous diverse fields of science and engineering. Some of the areas of applications of fractional differential equations include electrical network, viscoelasticity, optics, signal processing, finance, electromagnetics, control theory, acoustics, material science, nuclear reactor dynamics, biological systems and fluid mechanics, see in References [1–15]. Many processes in applied sciences and engineering can be modeled more accurately by fractional derivatives than integer order derivatives.…”

Section: Introductionmentioning

confidence: 99%

“…Various numerical schemes based on B‐spline collocation methods have been applied to solve a wide variety of problems, see [24–34]. Besides B‐spline collocation techniques, there are other methods that can be used to solve classic or fractional differential equations, such as finite difference methods [6, 7, 14], finite element method [35], boundary element methods [36, 37], meshless methods [38, 39] etc.…”

Section: Introductionmentioning

confidence: 99%

A high order numerical scheme for solving a class of non‐homogeneous time‐fractional reaction diffusion equation

Roul

Goura

2020

Numerical Methods Partial

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This paper is concerned with the development of a high order numerical technique for solving time‐fractional reaction–diffusion equation. The fractional derivative in the governing equation is described in the Caputo sense and a collocation method based on quintic B‐spline basis function is used to discretize the space variable. The stability and convergence analysis of the method are investigated, and it is shown that the proposed method converges to the exact solution of the problem with order of convergence O(Δt2 − α + Δx4), where α is the order of fractional derivative (0 < α < 1). Three test problems are considered to illustrate the efficiency and accuracy of present numerical scheme and to verify the theoretical result. It is shown that the order of fractional derivative has a profound effect on the solution of time‐fractional reaction–diffusion equation.

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Design and stability analysis of an implicit non-standard finite difference scheme for fractional neutron point kinetic equation

Cited by 29 publications

References 27 publications

Nonstandard finite difference schemes for general linear delay differential systems

Nonstandard finite difference schemes for general linear delay differential systems

Design and analysis of a high order computational technique for time‐fractional Black–Scholes model describing option pricing

A high order numerical scheme for solving a class of non‐homogeneous time‐fractional reaction diffusion equation

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