Computational Analysis of Fractional-Order KdV Systems in the Sense of the Caputo Operator via a Novel Transform (original) (raw)
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Axioms
This article investigates the seventh-order Lax’s Korteweg–de Vries equation using the Yang transform decomposition method (YTDM) and the homotopy perturbation transform method (HPTM). The physical phenomena that emerge in physics, engineering and chemistry are mathematically expressed by this equation. For instance, the KdV equation was constructed to represent a wide range of physical processes involving the evolution and interaction of nonlinear waves. In the Caputo sense, the fractional derivative is considered. We employed the Yang transform, the Adomian decomposition method and the homotopy perturbation method to obtain the solution to the time-fractional Lax’s Korteweg–de Vries problem. We examined and compared a particular example with the actual result to verify the approaches. By utilizing these methods, we can construct recurrence relations that represent the solution to the problem that is being proposed, and we are then able to present graphical representations that ena...
New Soliton Solutions of Time-Fractional Korteweg–de Vries Systems
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Model construction for different physical situations, and developing their solutions, are the major characteristics of the scientific work in physics and engineering. Korteweg–de Vries (KdV) models are very important due to their ability to capture different physical situations such as thin film flows and waves on shallow water surfaces. In this work, a new approach for predicting and analyzing nonlinear time-fractional coupled KdV systems is proposed based on Laplace transform and homotopy perturbation along with Caputo fractional derivatives. This algorithm provides a convergent series solution by applying simple steps through symbolic computations. The efficiency of the proposed algorithm is tested against different nonlinear time-fractional KdV systems, including dispersive long wave and generalized Hirota–Satsuma KdV systems. For validity purposes, the obtained results are compared with the existing solutions from the literature. The convergence of the proposed algorithm over t...
The Fractional Analysis of a Nonlinear mKdV Equation with Caputo Operator
Fractal and Fractional
In this study, we aim to provide reliable methods for the initial value problem of the fractional modified Korteweg–de Vries (mKdV) equations. Fractional differential equations are essential for more precise simulation of numerous processes. The hybrid Yang transformation decomposition method (YTDM) and Yang homotopy perturbation method (YHPM) are employed in a very simple and straightforward manner to handle the current problems. The derivative of fractional order is displayed in a Caputo form operator. To illustrate the conclusion given from the findings, a few numerical cases are taken into account for their approximate analytical solutions. We looked at two cases and contrasted them with the actual result to validate the methodologies. These techniques create recurrence relations representing the proposed problem’s solution. It is possible to find the series solutions to the given problems, and these solutions have components that converge to precise solutions more quickly. Tabl...
The Time-Fractional Coupled-Korteweg-de-Vries Equations
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We put into practice a relatively new analytical technique, the homotopy decomposition method, for solving the nonlinear fractional coupled-Korteweg-de-Vries equations. Numerical solutions are given, and some properties exhibit reasonable dependence on the fractional-order derivatives’ values. The fractional derivatives are described in the Caputo sense. The reliability of HDM and the reduction in computations give HDM a wider applicability. In addition, the calculations involved in HDM are very simple and straightforward. It is demonstrated that HDM is a powerful and efficient tool for FPDEs. It was also demonstrated that HDM is more efficient than the adomian decomposition method (ADM), variational iteration method (VIM), homotopy analysis method (HAM), and homotopy perturbation method (HPM).
Engineering Computations, 2019
Purpose This study aims to find the solution of time-fractional Korteweg–de-Vries (tfKdV) equations which may be used for modeling various wave phenomena using homotopy perturbation transform method (HPTM). Design/methodology/approach HPTM, which consists of mainly two parts, the first part is the application of Laplace transform to the differential equation and the second part is finding the convergent series-type solution using homotopy perturbation method (HPM), based on He’s polynomials. Findings The study obtained the solution of tfKdV equations. An existing result “as the fractional order of KdV equation given in the first example decreases the wave bifurcates into two peaks” is confirmed with present results by HPTM. A worth mentioning point may be noted from the results is that the number of terms required for acquiring the convergent solution may not be the same for different time-fractional orders. Originality/value Although third-order tfKdV and mKdV equations have alread...
A Novel Analytical View of Time-Fractional Korteweg-De Vries Equations via a New Integral Transform
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We put into practice relatively new analytical techniques, the Shehu decomposition method and the Shehu iterative transform method, for solving the nonlinear fractional coupled Korteweg-de Vries (KdV) equation. The KdV equation has been developed to represent a broad spectrum of physics behaviors of the evolution and association of nonlinear waves. Approximate-analytical solutions are presented in the form of a series with simple and straightforward components, and some aspects show an appropriate dependence on the values of the fractional-order derivatives that are, in a certain sense, symmetric. The fractional derivative is proposed in the Caputo sense. The uniqueness and convergence analysis is carried out. To comprehend the analytical procedure of both methods, three test examples are provided for the analytical results of the time-fractional KdV equation. Additionally, the efficiency of the mentioned procedures and the reduction in calculations provide broader applicability. It...
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This article applies the homotopy perturbation transform technique to analyze fractional-order nonlinear fifth-order Korteweg–de-Vries-type (KdV-type)/Kawahara-type equations. This method combines the Zain Ul Abadin Zafar-transform (ZZ-T) and the homotopy perturbation technique (HPT) to show the validation and efficiency of this technique to investigate three examples. It is also shown that the fractional and integer-order solutions have closed contact with the exact result. The suggested technique is found to be reliable, efficient, and straightforward to use for many related models of engineering and several branches of science, such as modeling nonlinear waves in different plasma models.
An explicit and numerical solutions of the fractional KdV equation
Mathematics and Computers in Simulation, 2005
In this paper, a fractional Korteweg-de Vries equation (KdV for short) with initial condition is introduced by replacing the first order time and space derivatives by fractional derivatives of order α and β with 0 < α, β ≤ 1, respectively. The fractional derivatives are described in the Caputo sense. The application of Adomian decomposition method, developed for differential equations of integer order, is extended to derive explicit and numerical solutions of the fractional KdV equation. The solutions of our model equation are calculated in the form of convergent series with easily computable components.
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In this article, we use the homotopy perturbation method and the Adomian decomposition method with the Yang transformation to discover analytical solution to the time-fractional coupled Schrödinger–KdV equation. In the Caputo sense, fractional derivatives are described. A convergent series is used to calculate the solutions of fractional PDEs. Analytical results achieved applying the homotopy perturbation and decomposition techniques are numerically calculated and represented in the form of tables and figures. The simplicity, efficacy, and high degree of accuracy of the used method are then demonstrated by comparing these solutions to the actual solutions and the results. Finally, the applied approaches are the most popular and convergent methods for solving nonlinear fractional-order partial deferential problems.
Analysis of a time fractional wave-like equation with the homotopy analysis method
Physics Letters A, 2008
The time fractional wave-like differential equation with a variable coefficient is studied analytically. By using a simple transformation, the governing equation is reduced to two fractional ordinary differential equations. Then the homotopy analysis method is employed to derive the solutions of these equations. The accurate series solutions are obtained. Especially, whenh f =h g = −1, these solutions are exactly the same as those results given by the Adomian decomposition method. The present work shows the validity and great potential of the homotopy analysis method for solving nonlinear fractional differential equations. The basic idea described in this Letter is expected to be further employed to solve other similar nonlinear problems in fractional calculus.