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Journal ArticleDOI

Numerical Solutions of Fractional Differential Equations by Using Laplace Transformation Method and Quadrature Rule

07 Sep 2021-Vol. 5, Iss: 3, pp 111
TL;DR: In this paper, a numerical scheme based on the numerical inversion of Laplace transform and equal width quadrature rule is proposed for solving a significant class of fractional differential equations.
Abstract: This paper introduces an efficient numerical scheme for solving a significant class of fractional differential equations. The major contributions made in this paper apply a direct approach based on a combination of time discretization and the Laplace transform method to transcribe the fractional differential problem under study into a dynamic linear equations system. The resulting problem is then solved by employing the numerical method of the quadrature rule, which is also a well-developed numerical method. The present numerical scheme, which is based on the numerical inversion of Laplace transform and equal-width quadrature rule is robust and efficient. Some numerical experiments are carried out to evaluate the performance and effectiveness of the suggested framework.
Citations
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TL;DR: In this article , the authors proposed an iterative method using Sawi transform to solve 1D, 2D, and 3D fractional hyperbolic telegraph equations in Caputo sense, which serve as a model for signal analysis of electrical impulse transmission and propagation.
Abstract: Abstract In the present study, 1D, 2D, and 3D fractional hyperbolic telegraph equations in Caputo sense have been solved using an iterative method using Sawi transform. These equations serve as a model for signal analysis of electrical impulse transmission and propagation. Along with a table of Sawi transform of some popular functions, some helpful results on Sawi transform are provided. To demonstrate the effectiveness of the suggested method, five examples in 1D, one example in 2D, and one example in 3D are solved using the proposed scheme. Error analysis comparing approximate and exact solutions using graphs and tables has been provided. The proposed scheme is robust, effective, and easy to implement and can be implemented on variety of fractional partial differential equations to obtain precise series approximations.
References
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Journal ArticleDOI
TL;DR: In this paper, the authors considered discretization in time of an inhomogeneous parabolic equation in a Banach space setting, using a representation of the solution as an integral along a smooth curve in the complex left half-plane which, after transformation to a finite interval, is then evaluated to high accuracy by a quadrature rule.
Abstract: We consider the discretization in time of an inhomogeneous parabolic equation in a Banach space setting, using a representation of the solution as an integral along a smooth curve in the complex left half-plane which, after transformation to a finite interval, is then evaluated to high accuracy by a quadrature rule. This reduces the problem to a finite set of elliptic equations with complex coefficients, which may be solved in parallel. The paper is a further development of earlier work by the authors, where we treated the homogeneous equation in a Hilbert space framework. Special attention is given here to the treatment of the forcing term. The method is combined with finite-element discretization in spatial variables.

126 citations

Journal ArticleDOI
TL;DR: McLean and Thomee as discussed by the authors studied three numerical methods for the discretization in time of a fractional-order evolution equation in a Banach space framework, each of which applied a quadrature rule to a contour integral representation of the solution.
Abstract: In a previous paper, McLean & Thomee (2009, J. Integr. Equ. Appl. (to appear)), we studied three numerical methods for the discretization in time of a fractional-order evolution equation in a Banach space framework. Each of the methods applied a quadrature rule to a contour integral representation of the solution in the complex plane, where for each quadrature point an elliptic boundary-value problem had to be solved to determine the value of the integrand. The first two methods involved the Laplace transform of the forcing term, but the third did not. We analysed both the quadrature error and the error arising from a spatial discretization by finite elements, measured in the L-2-norm. The present work extends our earlier results by proving error bounds in the technically more complicated case of the maximum norm. We also establish new regularity properties for the exact solution that are needed for our analysis.

66 citations

Journal ArticleDOI
TL;DR: In this paper, a comparison of the efficiency of three alternative decomposition schemes for the approximate solution of multi-term fractional differential equations using the Caputo form of the fractional derivative is given.

57 citations

Journal ArticleDOI
TL;DR: This study proposes a new approach for the optimization of phase and magnitude responses of fractional-order capacitive and inductive elements based on the mixed integer-order genetic algorithm (GA), over a bandwidth of four-decade, and operating up to 1 GHz with a low phase error.
Abstract: This study proposes a new approach for the optimization of phase and magnitude responses of fractional-order capacitive and inductive elements based on the mixed integer-order genetic algorithm (GA), over a bandwidth of four-decade, and operating up to 1 GHz with a low phase error of approximately ±1°. It provides a phase optimization in the desired bandwidth with minimal branch number and avoids the use of negative component values, and any complex mathematical analysis. Standardized, IEC 60063 compliant commercially available passive component values are used; hence, no correction on passive elements is required. To the best knowledge of the authors, this approach is proposed for the first time in the literature. As validation, we present numerical simulations using MATLAB ® and experimental measurement results, in particular, the Foster-II and Valsa structures with five branches for precise and/or high-frequency applications. Indeed, the results demonstrate excellent performance and significant improvements over the Oustaloup approximation, the Valsa recursive algorithm, and the continued fraction expansion and the adaptability of the GA-based design with five different types of distributed RC/RL network.

52 citations

Journal ArticleDOI
TL;DR: To control the deviation in frequency and power, an integration in the environment of fractional order (FO) calculus for proportional–integral–derivative (PID) controller and fuzzy controller, termed with FO-Fuzzy PID controller tuned with quasi-opposition based harmonic search (QOHS) algorithm has been proposed.

48 citations