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

Liaqat, Muhammad Imran; Khan, Adnan; Akgül, Ali; Ali, Md. Shajib

doi:10.1155/2022/6333084

Cited by 6 publications

(3 citation statements)

References 60 publications

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“…Researchers have used a variety of transforms along with well-known methodologies to solve differential equations, including non-integer order logistic differential models [ 19 ], non-integer order BBM-Burger equations [ 20 ], non-integer order relaxation-oscillation differential equations [ 21 ], and more. The Elzaki transform with residual power series method (ERPSM) has been successfully used to solve some well-known differential equations of non-integer order.…”

Section: Introductionmentioning

confidence: 99%

Elzaki residual power series method to solve fractional diffusion equation

Pant,

Arora,

Emadifar

2024

PLoS ONE

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The time-fractional order differential equations are used in many different contexts to analyse the integrated scientific phenomenon. Hence these equations are the point of interest of the researchers. In this work, the diffusion equation for a one-dimensional time-fractional order is solved using a combination of residual power series method with Elzaki transforms. The residual power series approach is a useful technique for finding approximate analytical solutions of fractional differential equations that needs the residual function’s (n-1)α derivative. Since it is challenging to determine a function’s fractional-order derivative, the traditional residual power series method’s application is somewhat constrained. The Elzaki transform with residual power series method is an attempt to get over the limitations of the residual power series method. The obtained numerical solutions are compared with the exact solution of this equation to discuss the method’s applicability and efficiency. The results are also graphically displayed to show how the fractional derivative influences the behaviour of the solutions to the suggested method.

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Section: Introductionmentioning

confidence: 99%

Elzaki residual power series method to solve fractional diffusion equation

Pant,

Arora,

Emadifar

2024

PLoS ONE

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“…Although fractional derivatives can be defned in a variety of ways, not all of them are generally used. Te Atangana-Baleanu, Riemann-Liouville (R-L), Caputo-Fabrizio, Caputo, and conformable operators are the most frequently used [6][7][8][9][10][11][12]. In some cases, fractional derivatives are preferable to integerorder derivatives for modeling because they can simulate and analyze complicated systems having complicated nonlinear processes and higher-order behaviors.…”

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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“…As a result of the foregoing, researchers have devised a variety of numerical strategies for solving nonlinear FODEs. A few examples include the Elzaki residual power series method [28], the Haar Wavelet method [29], the operational Matrix Technique [30], the reduced differential transform method [31], the spectral Tau approach [32], the reproducing kernel technique [33], and the fractional power series technique [34].…”

Section: Introductionmentioning

confidence: 99%

The Fractional Series Solutions for the Conformable Time‐Fractional Swift‐Hohenberg Equation through the Conformable Shehu Daftardar‐Jafari Approach with Comparative Analysis

Liaqat

Okyere

2022

Journal of Mathematics

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The major objective of this study is to derive fractional series solutions of the time-fractional Swift-Hohenberg equations (TFSHEs) in the sense of conformable derivative using the conformable Shehu transform (CST) and the Daftardar-Jafari approach (DJA). We call it the conformable Shehu Daftardar-Jafari approach (CSDJA). One of the universal equations used in the description of pattern formation in spatially extended dissipative systems is the Swift-Hohenberg equation. To assess the effectiveness and consistency of the suggested approach, the numerical results are compared with those obtained by the Elzaki decomposition method (EDM) in the sense of relative and absolute error functions, proving that the CSDJA is an effective substitute for techniques that use He's or Adomian polynomials to solve TFSHEs. The transition from the imprecise solution to the precise solution at various values of fractional-order derivatives is shown using the recurrence error function. Furthermore, the exact and approximative solutions are compared using 2D and 3D graphics and also numerically in the form of relative and absolute error functions. The results show that the procedure is quick, precise, and easy to implement, and it yields outstanding results. The recommended approach's strength, which gives it an advantage over the Adomian decomposition and homotopy perturbation methods, is its algorithm for dealing with nonlinear problems without the use of Adomian polynomials or He's polynomials. The advantage of this method is that it does not make any assumptions about physical parameters. As a result, it can be used to solve both weakly and strongly nonlinear problems and circumvent some of the drawbacks of perturbation techniques.

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

Cited by 6 publications

References 60 publications

Elzaki residual power series method to solve fractional diffusion equation

Elzaki residual power series method to solve fractional diffusion equation

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

The Fractional Series Solutions for the Conformable Time‐Fractional Swift‐Hohenberg Equation through the Conformable Shehu Daftardar‐Jafari Approach with Comparative Analysis

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