scholarly journals Combination of Laplace transform and residual power series techniques to solve autonomous n-dimensional fractional nonlinear systems

2021 ◽  
Vol 10 (1) ◽  
pp. 282-292
Author(s):  
Marwan Alquran ◽  
Maysa Alsukhour ◽  
Mohammed Ali ◽  
Imad Jaradat
Keyword(s):  
Nonlinear Systems ◽  
Power Series ◽  
Laplace Transform ◽  
Iterative Algorithm ◽  
Autonomous Systems ◽  
Numerical Examples ◽  
The Laplace Transform ◽  
Residual Power Series ◽  
The Impact ◽  
Residual Power

Abstract In this work, a new iterative algorithm is presented to solve autonomous n-dimensional fractional nonlinear systems analytically. The suggested scheme is combination of two methods; the Laplace transform and the residual power series. The methodology of this algorithm is presented in details. For the accuracy and effectiveness purposes, two numerical examples are discussed. Finally, the impact of the fractional order acting on these autonomous systems is investigated using graphs and tables.

scholarly journals A new iterative algorithm on the time-fractional Fisher equation: Residual power series method

2017 ◽  
Vol 9 (9) ◽  
pp. 168781401771600 ◽  
Author(s):  
Maysaa’ Mohamed Al Qurashi ◽  
Zeliha Korpinar ◽  
Dumitru Baleanu ◽  
Mustafa Inc
Keyword(s):  
Power Series ◽  
Iterative Algorithm ◽  
Fisher Equation ◽  
Series Method ◽  
Power Series Method ◽  
Residual Power Series Method ◽  
Residual Power Series ◽  
Residual Power

scholarly journals An Attractive Approach Associated with Transform Functions for Solving Certain Fractional Swift-Hohenberg Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Mohammad Alaroud ◽  
Nedal Tahat ◽  
Shrideh Al-Omari ◽  
D. L. Suthar ◽  
Selma Gulyaz-Ozyurt
Keyword(s):  
Approximate Solutions ◽  
Caputo Fractional Derivatives ◽  
Approximate Analytical Solutions ◽  
The Laplace Transform ◽  
Fractional Models ◽  
Residual Power Series ◽  
Linear And Nonlinear ◽  
The Impact ◽  
Generalized Taylor’S Formula ◽  
Residual Power

Many phenomena in physics and engineering can be built by linear and nonlinear fractional partial differential equations which are considered an accurate instrument to interpret these phenomena. In the current manuscript, the approximate analytical solutions for linear and nonlinear time-fractional Swift-Hohenberg equations are created and studied by means of a recent superb technique, named the Laplace residual power series (LRPS) technique under the time-Caputo fractional derivatives. The proposed technique is a combination of the generalized Taylor’s formula and the Laplace transform operator, which depends mainly on the concept of limit at infinity to find the unknown functions for the fractional series expansions in the Laplace space with fewer computations and more accuracy comparing with the classical RPS that depends on the Caputo fractional derivative for each step in obtaining the coefficient expansion. To test the simplicity, performance, and applicability of the present method, three numerical problems of the time-fractional Swift-Hohenberg initial value problems are considered. The impact of the fractional order β on the behavior of the approximate solutions at fixed bifurcation parameter is shown graphically and numerically. Obtained results emphasized that the LRPS technique is an easy, efficient, and speed approach for the exact description of the linear and nonlinear time-fractional models that arise in natural sciences.

scholarly journals Adaptation of Residual-Error Series Algorithm to Handle Fractional System of Partial Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2868
Author(s):  
Hussam Aljarrah ◽  
Mohammad Alaroud ◽  
Anuar Ishak ◽  
Maslina Darus
Keyword(s):  
Partial Differential Equations ◽  
Power Series ◽  
Differential Equations ◽  
Applied Mathematics ◽  
Approximate Solutions ◽  
Partial Differential ◽  
Adomian Decomposition ◽  
Residual Power Series ◽  
The Impact ◽  
Residual Power

In this article, an attractive numeric–analytic algorithm, called the fractional residual power series algorithm, is implemented for predicting the approximate solutions for a certain class of fractional systems of partial differential equations in terms of Caputo fractional differentiability. The solution methodology combines the residual function and the fractional Taylor’s formula. In this context, the proposed algorithm provides the unknown coefficients of the expansion series for the governed system by a straightforward pattern as well as it presents the solutions in a systematic manner without including any restrictive conditions. To enhance the theoretical framework, some numerical examples are tested and discussed to detect the simplicity, performance, and applicability of the proposed algorithm. Numerical simulations and graphical plots are provided to check the impact of the fractional order on the geometric behavior of the fractional residual power series solutions. Moreover, the efficiency of this algorithm is discussed by comparing the obtained results with other existing methods such as Laplace Adomian decomposition and Iterative methods. Simulation of the results shows that the fractional residual power series technique is an accurate and very attractive tool to obtain the solutions for nonlinear fractional partial differential equations that occur in applied mathematics, physics, and engineering.

scholarly journals Approximating the Laplace transform of the sum of dependent lognormals

2016 ◽  
Vol 48 (A) ◽  
pp. 203-215 ◽  
Author(s):  
Patrick J. Laub ◽  
Søren Asmussen ◽  
Jens L. Jensen ◽  
Leonardo Rojas-Nandayapa
Keyword(s):  
Monte Carlo ◽  
Laplace Transform ◽  
Taylor Expansion ◽  
Multivariate Normal ◽  
Numerical Examples ◽  
Quasi Monte Carlo ◽  
Error Factor ◽  
The Laplace Transform ◽  
Laplace Transform Inversion ◽  
Mean Vector

AbstractLet (X1,...,Xn) be multivariate normal, with mean vector 𝛍 and covariance matrix 𝚺, and letSn=eX1+⋯+eXn. The Laplace transform ℒ(θ)=𝔼e-θSn∝∫exp{-hθ(𝒙)}d𝒙 is represented as ℒ̃(θ)I(θ), where ℒ̃(θ) is given in closed form andI(θ) is the error factor (≈1). We obtain ℒ̃(θ) by replacinghθ(𝒙) with a second-order Taylor expansion around its minimiser 𝒙*. An algorithm for calculating the asymptotic expansion of 𝒙*is presented, and it is shown thatI(θ)→ 1 as θ→∞. A variety of numerical methods for evaluatingI(θ) is discussed, including Monte Carlo with importance sampling and quasi-Monte Carlo. Numerical examples (including Laplace-transform inversion for the density ofSn) are also given.

scholarly journals Adaptation of Conformable Residual Power Series Scheme in Solving Nonlinear Fractional Quantum Mechanics Problems

2020 ◽  
Vol 10 (3) ◽  
pp. 890 ◽  
Author(s):  
Mohammed Shqair ◽  
Mohammed Al-Smadi ◽  
Shaher Momani ◽  
Essam El-Zahar
Keyword(s):  
Quantum Mechanics ◽  
Power Series ◽  
Quantum Physics ◽  
Taylor Series Expansion ◽  
Fractional Quantum ◽  
Analytical Technique ◽  
Different Types ◽  
Residual Power Series ◽  
Residual Errors ◽  
Residual Power

In this paper, the general state of quantum mechanics equations that can be typically expressed by nonlinear fractional Schrödinger models will be solved based on an attractive efficient analytical technique, namely the conformable residual power series (CRPS). The fractional derivative is considered in a conformable sense. The desired analytical solution is obtained using conformable Taylor series expansion through substituting a truncated conformable fractional series and minimizing its residual errors to extract a supportive approximate solution in a rapidly convergent fractional series. This adaptation can be implemented as a novel alternative technique to deal with many nonlinear issues occurring in quantum physics. The effectiveness and feasibility of the CRPS procedures are illustrated by verifying three realistic applications. The obtained numerical results and graphical consequences indicate that the suggested method is a convenient and remarkably powerful tool in solving different types of fractional partial differential models.

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