A Highly Accurate Technique to Obtain Exact Solutions to Time‐Fractional Quantum Mechanics Problems with Zero and Nonzero Trapping Potential

Liaqat, Muhammad Imran; Khan, Adnan; Alam, Md. Ashraful; Pandit, M. K.

doi:10.1155/2022/9999070

Cited by 9 publications

(4 citation statements)

References 41 publications

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“…Under the specifed initial and boundary conditions, numerical methods present a potent alternative tool for solving FODEs. Several numerical techniques, including the Shehu decomposition approach [13], the diferential transform method [14], the variational iteration method [15], the operational matrix approach [16], the homotopy analysis technique [17], the Aboodh transform decomposition method [18], the fnite diference method [19], the fractional power series method [20], the Chebyshev polynomials method [21], the residual power series method [22], and the natural transform homotopy perturbation method [23], have been developed in recent years for solving FODEs.…”

Section: Introductionmentioning

confidence: 99%

Comparative Analysis of the Time-Fractional Black–Scholes Option Pricing Equations (BSOPE) by the Laplace Residual Power Series Method (LRPSM)

Liaqat

Okyere

2023

Journal of Mathematics

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The residual power series method is effective for obtaining solutions to fractional-order differential equations. However, the procedure needs the n − 1 ϖ derivative of the residual function. We are all aware of the difficulty of computing the fractional derivative of a function. In this article, we considered the simple and efficient method known as the Laplace residual power series method (LRPSM) to find the analytical approximate and exact solutions of the time-fractional Black–Scholes option pricing equations (BSOPE) in the sense of the Caputo derivative. This approach combines the Laplace transform and the residual power series method. The suggested method just needs the idea of an infinite limit, so the computations required to determine the coefficients are minimal. The obtained results are compared in the sense of absolute errors against those of other approaches, such as the homotopy perturbation method, the Aboodh transform decomposition method, and the projected differential transform method. The results obtained using the provided method show strong agreement with different series solution methods, demonstrating that the suggested method is a suitable alternative tool to the methods based on He’s or Adomian polynomials.

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Section: Introductionmentioning

confidence: 99%

Comparative Analysis of the Time-Fractional Black–Scholes Option Pricing Equations (BSOPE) by the Laplace Residual Power Series Method (LRPSM)

Liaqat

Okyere

2023

Journal of Mathematics

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“…Exploring the analytic solution of FDEs and FIDEs is difficult in most cases, even though abundant efforts have been introduced recently to develop emerging numerical and approximate-analytical techniques for finding out the solutions to linear and nonlinear fractional problems. Among these methods, the kernel Hilbert space method [29], the Haar wavelet method [30], the Adomian decomposition method [31], the homotopy analysis method [32], the finite difference method [33], the Taylor series expansion method [34], the collocation method [35], the Aboodh transform decomposition method [36], and the residual fractional power series (FPS) method [37,38] have been reproduced. The FPS method is one of the semianalytical techniques which befits both linear and nonlinear FDEs [39][40][41].…”

Section: Introductionmentioning

confidence: 99%

Extended Laplace Power Series Method for Solving Nonlinear Caputo Fractional Volterra Integro-Differential Equations

et al. 2023

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In this paper, we compile the fractional power series method and the Laplace transform to design a new algorithm for solving the fractional Volterra integro-differential equation. For that, we assume the Laplace power series (LPS) solution in terms of power q=1m,m∈Z+, where the fractional derivative of order α=qγ, for which γ∈Z+. This assumption will help us to write the integral, the kernel, and the nonhomogeneous terms as a LPS with the same power. The recurrence relations for finding the series coefficients can be constructed using this form. To demonstrate the algorithm’s accuracy, the residual error is defined and calculated for several values of the fractional derivative. Two strongly nonlinear examples are discussed to provide the efficiency of the algorithm. The algorithm gains powerful results for this kind of fractional problem. Under Caputo meaning of the symmetry order, the obtained results are illustrated numerically and graphically. Geometrically, the behavior of the obtained solutions declares that the changing of the fractional derivative parameter values in their domain alters the style of these solutions in a symmetric meaning, as well as indicates harmony and symmetry, which leads them to fully coincide at the value of the ordinary derivative. From these simulations, the results report that the recommended novel algorithm is a straightforward, accurate, and superb tool to generate analytic-approximate solutions for integral and integro-differential equations of fractional order.

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“…In recent years, researchers have proposed lots of new different transformations to solve a variety of mathematical problems. FODEs are solved using the Aboodh transform [25], fractional complex transform [26], travelling wave transform [27], Sumudu transform [28], and ZZ transform [29]. These transformations are paired with additional analytical, numerical, or homotopy-based techniques to handle FODEs [30][31][32][33][34][35].…”

Section: Introductionmentioning

confidence: 99%

A Novel Numerical Technique for Fractional Ordinary Differential Equations with Proportional Delay

Liaqat

Khan

Akgül

et al. 2022

Journal of Function Spaces

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Some researchers have combined two powerful techniques to establish a new method for solving fractional-order differential equations. In this study, we used a new combined technique, known as the Elzaki residual power series method (ERPSM), to offer approximate and exact solutions for fractional multipantograph systems (FMPS) and pantograph differential equations (PDEs). In Caputo logic, the fractional-order derivative operator is measured. The Elzaki transform method and the residual power series method (RPSM) are combined in this novel technique. The suggested technique is based on a new version of Taylor’s series that generates a convergent series as a solution. Establishing the coefficients for a series, like the RPSM, necessitates computing the fractional derivatives each time. As ERPSM just requires the concept of a zero limit, we simply need a few computations to get the coefficients. The novel technique solves nonlinear problems without the need for He’s and Adomian polynomials, which is an advantage over the other combined methods based on homotopy perturbation and Adomian decomposition methods. The relative, recurrence, and absolute errors of the problems are analyzed to evaluate the efficiency and consistency of the presented method. Graphical significances are also identified for various values of fractional-order derivatives. As a result, the procedure is quick, precise, and easy to implement, and it yields outstanding results.

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A Highly Accurate Technique to Obtain Exact Solutions to Time‐Fractional Quantum Mechanics Problems with Zero and Nonzero Trapping Potential

Cited by 9 publications

References 41 publications

Comparative Analysis of the Time-Fractional Black–Scholes Option Pricing Equations (BSOPE) by the Laplace Residual Power Series Method (LRPSM)

Comparative Analysis of the Time-Fractional Black–Scholes Option Pricing Equations (BSOPE) by the Laplace Residual Power Series Method (LRPSM)

Extended Laplace Power Series Method for Solving Nonlinear Caputo Fractional Volterra Integro-Differential Equations

A Novel Numerical Technique for Fractional Ordinary Differential Equations with Proportional Delay

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