An exponential B-spline collocation method for the fractional sub-diffusion equation

Zhu, Xiaogang; Nie, Yufeng; Yuan, Zhanbin; Wang, Jun Gang; Yang, Zongze

doi:10.1186/s13662-017-1328-6

Cited by 13 publications

(9 citation statements)

References 37 publications

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“…McCartin also showed that the exponential splines accept a basis of B-splines. They are used in approximating the solutions of various classes of problems in differential equations [38][39][40][41].…”

Section: Introductionmentioning

confidence: 99%

An efficient numerical method based on exponential B-splines for time-fractional Black-Scholes equation governing European options

Singh¹,

Kumar²

2022

Preprint

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In this paper a time-fractional Black-Scholes model (TFBSM) is considered to study the price change of the underlying fractal transmission system. We develop and analyze a numerical method to solve the TFBSM governing European options. The numerical method combines the exponential B-spline collocation to discretize in space and a finite difference method to discretize in time. The method is shown to be unconditionally stable using von-Neumann analysis. Also, the method is proved to be convergent of order two in space and 2 − µ is time, where µ is order of the fractional derivative. We implement the method on various numerical examples in order to illustrate the accuracy of the method, and validation of the theoretical findings. In addition, as an application, the method is used to price several different European options such as the European call option, European put option, and European double barrier knock-out call option.

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Section: Introductionmentioning

confidence: 99%

An efficient numerical method based on exponential B-splines for time-fractional Black-Scholes equation governing European options

Singh¹,

Kumar²

2022

Preprint

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“…Several differential equations have been solved by utilizing this method. [24][25][26][27][28][29][30] Using this technique, one can avail of the solution even between nodes, which is an advantage over the finite difference scheme. There is no requirement to compute quadratures as in the finite element technique.…”

Section: Introductionmentioning

confidence: 99%

“…The exponential B‐spline technique is based on piece‐wise non‐polynomial functions and was developed by McCartin, 23 and it is the generalization of the cubic spline. Several differential equations have been solved by utilizing this method 24–30 . Using this technique, one can avail of the solution even between nodes, which is an advantage over the finite difference scheme.…”

Section: Introductionmentioning

confidence: 99%

A computational procedure and analysis for multi‐term time‐fractional Burgers‐type equation

Kanth

Garg

2022

Math Methods in App Sciences

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This paper presents a new numerical algorithm dealing with multi-term time-fractional Burgers-type equation involving the Caputo derivative. The proposed method consists of temporal discretization of L2 formula and spatial discretization using the exponential B-splines. The semi implicit approach is applied to discretize the nonlinear term u𝜕 x u. We adopt the Von-Neumann method to study stability. We also establish the convergence analysis. The proposed method is employed to solve a few numerical examples in order to test its efficiency and accuracy. Comparisons with the recent works confirm the efficiency and robustness of the proposed method.

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“…Zahra and Elkholy [13] use cubic splines to represent a numerical solution of fractional differential equations. Nonclassical diffusion problems by Ismail et al [14], advection-diffusion problems by Nazir et al [15], and fractional subdiffusion equation by Zhu et al [16] are solved by using B-spline collocation methods. In recent times, Hashmi et al [17,18] has solved Hunter Saxton equation and space fractional PDE by cubic trigonometric and hybrid B-spline method.…”

Section: Introductionmentioning

confidence: 99%

A New Variant of B-Spline for the Solution of Modified Fractional Anomalous Subdiffusion Equation

Hashmi

Shehzad

Ashraf

et al. 2021

Journal of Function Spaces

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The objective of this paper is to present an efficient numerical technique for solving time fractional modified anomalous subdiffusion equation. Anomalous diffusion equation has its role in various branches of biological sciences. B-spline is a piecewise function to draw curves and surfaces, which maintain its degree of smoothness at the connecting points. B-spline provides an active process of approximation to the limit curve. In current attempt, B-spline curve is used to approximate the solution curve of time fractional modified anomalous subdiffusion equation. The process is kept simple involving collocation procedure to the data points. The time fractional derivative is approximated with the discretized form of the Riemann-Liouville derivative. The process results in the form of system of algebraic equations, which is solved using a variant of Thomas algorithm. In order to ensure the convergence of the procedure, a valid method named Von Neumann stability analysis is attempted. The graphical and tabular display of results for the illustrated examples is presented, which stamped the efficiency of the proposed algorithm.

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An exponential B-spline collocation method for the fractional sub-diffusion equation

Cited by 13 publications

References 37 publications

An efficient numerical method based on exponential B-splines for time-fractional Black-Scholes equation governing European options

An efficient numerical method based on exponential B-splines for time-fractional Black-Scholes equation governing European options

A computational procedure and analysis for multi‐term time‐fractional Burgers‐type equation

A New Variant of B-Spline for the Solution of Modified Fractional Anomalous Subdiffusion Equation

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