Nonlinear Time-fractional Partial Differential Equations Research Articles

Solving Nonlinear Time-Fractional Partial Differential Equations Using Conformable Fractional Reduced Differential Transform with Adomian Decomposition Method

In this article, we use a new technique called conformable fractional reduced differential transform (CFRDT) with Adomian decomposition to estimate the solution of one and two-dimensional time-fractional partial linear and nonlinear differential equations with initial values. We explain the convergence analysis of this technique. The obtained results illustrate that the novel method is efficient and easy to use to find approximate solutions for the time-fractional partial differential equations (PDEs). Thus, the suggested method has a significant impact on how engineering, physics, and other disciplines solve fractional PDEs. Furthermore, we analyze the solution of problems with a 2D or 3D graphical representation by using Mathematica software.

A comparative analytical investigation for some linear and nonlinear time-fractional partial differential equations in the framework of the Aboodh transformation

This article discusses two simple, complication-free, and effective methods for solving fractional-order linear and nonlinear partial differential equations analytically: the Aboodh residual power series method (ARPSM) and the Aboodh transform iteration method (ATIM). The Caputo operator is utilized to define fractional order derivatives. In these methods, the analytical approximations are derived in series form. We calculate the first terms of the series and then estimate the absolute error resulting from leaving out the remaining terms to ensure the accuracy of the derived approximations and determine the accuracy and efficiency of the suggested methods. The derived approximations are discussed numerically using some values for the relevant parameters to the subject of the study. Useful examples are thought to illustrate the practical application of current approaches. We also examine the fractional order results that converge to the integer order solutions to ensure the accuracy of the derived approximations. Many researchers, particularly those in plasma physics, are anticipated to gain from modeling evolution equations describing nonlinear events in plasma systems.

Space-Time Continuous and Time Discontinuous Galerkin Schemes Based on Isogeometric Analysis for Nonlinear Time-Fractional Partial Differential Equations

Space-Time Continuous and Time Discontinuous Galerkin Schemes Based on Isogeometric Analysis for Nonlinear Time-Fractional Partial Differential Equations

An efficient method for the analytical study of linear and nonlinear time-fractional partial differential equations with variable coefficients

Метод остаточных степенных рядов эффективен для получения приближенных аналитических решений дифференциальных уравнений дробного порядка. Вычисление дробной производной для коэффициентов степенного ряда, аппроксимирующего точное решение дифференциального уравнения, является недостатком этого метода. Другие известные методы приближенного интегрирования, такие как гомотопическое возмущение, разложение Адомиана и методы вариационных итераций, основываются на интегрировании для получения степенного ряда. Известна сложность вычисления дробных производных и интегрирования функций при построении степенного ряда для решения уравнений математической физики дробного порядка, поэтому использование упомянутых выше методов ограничено спецификой решаемой задачи. В настоящей статье получены приближенные и точные аналитические решения уравнений в частных производных переменными коэффициентами при использовании метода рядов остаточных степеней Лапласа в смысле дробной производной Герасимова-Капуто для времени. Этот метод помог преодолеть ограничения упомянутых выше способов интегрирования уравнений дробного порядка. Метод остаточных степенных рядов Лапласа лучше использовать при вычислении коэффициентов членов в решении ряда, применяя принцип прямого предела на бесконечности. Он также более эффективен, чем различные методы решения, если не использовать полиномы Адомиана и He для решения нелинейных задач дробного порядка. В статье исследуются относительные, повторяющиеся и абсолютные ошибки для трех задач математической физики для оценки достоверности предложенного метода. Результаты показывают, что сконструированный метод является альтернативой различным методам для построения решения рядами при решении уравнений в частных производных с дробным временем.

A new analytical approximate solution of fractional coupled Korteweg-de Vries system

The main objective of this work is to present a modification of the Mittag- Leffler function to deduce a relatively new analytical approximate method (for short MMLFM) able to solve time-fractional nonlinear partial differential equations (PDEs). Moreover, we employ the MMLFM to solve the time-fractional coupled Korteweg?de Vries (KdV) model described by two nonlinear fractional partial differential equations (FPDEs) based upon Caputo fractional derivative (CFD). The simulation of projected results is presented in some figures and tables. Furthermore, we compare our solutions when ? = 1 with known exact solutions which indicate a good agreement, in addition, we compare our outcomes with the results obtained by other methods in the literature such as the Natural decomposing method (NDM) and homotopy decomposition method (HDM) in order to prove the reliability and efficiency of our used method. Also, we display solutions with different values of ? to present the effect of the fractional order on the proposed problem. The results of this article reveal the advantages of the MMLFM, which is simple, reliable, accurate, needs simple mathematical computations, is rapidly convergent to the exact solution, have a straightforward and easy algorithm compared to other analytical methods to study linear and nonlinear FPDEs, which makes this technique suited for real industrial or medical applications.

A novel Elzaki transform homotopy perturbation method for solving time-fractional non-linear partial differential equations

This paper presents the solution of important types of non-linear time-fractional partial differential equations via the conformable Elzaki transform Homotopy perturbation method. We apply the proposed technique to solve four types of non-linear time-fractional partial differential equations. In addition, we establish the results on the uniqueness and convergence of the solution. Finally, the numerical results for a variety of α values are briefly examined. The proposed method performs well in terms of simplicity and efficiency.

Novel Approaches for Solving Fuzzy Fractional Partial Differential Equations

In this paper, we present a comparison of several important methods to solve fuzzy partial differential equations (PDEs). These methods include the fuzzy reduced differential transform method (RDTM), fuzzy Adomian decomposition method (ADM), fuzzy Homotopy perturbation method (HPM), and fuzzy Homotopy analysis method (HAM). A distinguishing practical feature of these techniques is administered without the need to use discretion or restricted assumptions. Moreover, we investigate the fuzzy (n+1)-dimensional fractional RDTM to obtain the solutions of fuzzy fractional PDEs. The much more distinctive element of this method is that it requires no predetermined assumptions, and reduces the computational effort. We apply the suggested techniques to a set of initial valued problems and get approximate numerical solutions for linear and nonlinear time-fractional PDEs. It is demonstrated that the fuzzy (n+1)-dimensional fractional RDTM is both accurate and simple to use. The methods are based on gH-differentiability and fuzzy fractional derivatives. Some illustrative numerical examples are given to demonstrate the effectiveness of our proposed methods. The results show that the methods are powerful mathematical tools for solving fuzzy partial differential equations.

Adaptive multi-step piecewise interpolation reproducing kernel method for solving the nonlinear time-fractional partial differential equation arising from financial economics

This paper is aimed at solving the nonlinear time-fractional partial differential equation with two small parameters arising from option pricing model in financial economics. The traditional reproducing kernel (RK) method which deals with this problem is very troublesome. This paper proposes a new method by adaptive multi-step piecewise interpolation reproducing kernel (AMPIRK) method for the first time. This method has three obvious advantages which are as follows. Firstly, the piecewise number is reduced. Secondly, the calculation accuracy is improved. Finally, the waste time caused by too many fragments is avoided. Then four numerical examples show that this new method has a higher precision and it is a more timesaving numerical method than the others. The research in this paper provides a powerful mathematical tool for solving time-fractional option pricing model which will play an important role in financial economics.

Homotopi pertürbasyon Elzaki dönüşümü yöntemi ile doğrusal olmayan zaman-kesirli kısmi diferansiyel denklemler için yeni yaklaşık analitik çözümler

Some nonlinear time-fractional partial differential equations are solved by homotopy perturbation Elzaki transform method. The fractional derivatives are defined in the Caputo sense. The applications are examined by homotopy perturbation Elzaki transform method. Besides, the graphs of the solutions are plotted in the MAPLE software. Also, absolute error comparison of homotopy perturbation Elzaki transform method and homotopy perturbation Sumudu transform method solutions with the exact solution of nonlinear time-fractional partial differential equations is presented. In addition, this absolute error comparison is indicated in the tables. The novelty of this article is the first analysis of both the gas dynamics equation of Caputo fractional order and the Klein-Gordon equation of Caputo fractional order via this method. Thus, homotopy perturbation Elzaki transform method is quick and effective in obtaining the analytical solutions of time-fractional partial differential equations.

The Caputo–Fabrizio time-fractional Sharma–Tasso–Olver–Burgers equation and its valid approximations

Studying the dynamics of solitons in nonlinear time-fractional partial differential equations has received substantial attention, in the last decades. The main aim of the current investigation is to consider the time-fractional Sharma–Tasso–Olver–Burgers (STOB) equation in the Caputo–Fabrizio (CF) context and obtain its valid approximations through adopting a mixed approach composed of the homotopy analysis method (HAM) and the Laplace transform. The existence and uniqueness of the solution of the time-fractional STOB equation in the CF context are investigated by demonstrating the Lipschitz condition for as the kernel and giving some theorems. To illustrate the CF operator effect on the dynamics of the obtained solitons, several two- and three-dimensional plots are formally considered. It is shown that the mixed approach is capable of producing valid approximations to the time-fractional STOB equation in the CF context.

Analytical approximate solutions of fractional nonlinear Drinfeld–Sokolov–Wilson model using modified Mittag–Leffler function

In this article, we introduce a modification of the generalized Mittag–Leffler function method (GMLFM) in order to provide appropriate analytical-approximate solutions for time-fractional nonlinear partial differential equations (PDEs). Moreover, we implement the proposed method on the Drinfeld–Sokolov–Wilson (DSW) model which is considered one of the most important mathematical models describing mathematical physics and dispersive water waves. We use Caputo fractional derivative (CFD) as a time-fractional operator for the mathematical formulation of this model. We introduce the results obtained by modified generalized Mittag Leffler function method (MGMLFM) when the fractional operator takes different values to show the impact of fractional order on the solution of the fractional DSW model, as well, we compare our results when α=1 with the exact solution to prove the efficiency of the used method. Furthermore, a simulation of our outcomes in some graphical and tabular forms is presented to be clearer for the reader. This simulation detects that by decreasing the value of time t the obtained solutions of the DSW system are remarkably close to the given exact solutions. Furthermore, decreasing the fractional operator from α=1 rapidly increases the amplitude of the solitary wave of the model. Also, results reveal that the absolute error decreases as spatial variable x increases at small-time t. The outcomes of this article confirm that this modification made to the GMLFM leads us to an effective, easy and accurate method to solve linear and nonlinear fractional PDEs which need little effort for computations.

Invariant subspace method for [formula omitted]-dimensional non-linear time-fractional partial differential equations

In this paper, we generalize the theory of the invariant subspace method to (m+1)-dimensional non-linear time-fractional partial differential equations for the first time. More specifically, the applicability and efficacy of the method have been systematically investigated through the (3+1)-dimensional generalized non-linear time-fractional diffusion–convection-wave equation along with appropriate initial conditions. This systematic investigation provides an important technique for finding a large class of various types of invariant subspaces with different dimensions for the above-mentioned equation. Additionally, we have shown that the obtained invariant subspaces help to derive a variety of exact solutions that can be expressed as the combinations of exponential, trigonometric, polynomial and well-known Mittag-Leffler functions.

Efficient Approaches for Solving Systems of Nonlinear Time-Fractional Partial Differential Equations

In this work, we present a modified generalized Mittag–Leffler function method (MGMLFM) and Laplace Adomian decomposition method (LADM) to get an analytic-approximate solution for nonlinear systems of partial differential equations (PDEs) of fractional-order in the Caputo derivative. We apply the MGMLFM and LADM on systems of nonlinear time-fractional PDEs. Precisely, we consider some important fractional-order nonlinear systems, namely Broer–Kaup (BK) and Burgers, which have found major significance because they arise in many physical applications such as shock wave, wave processes, vorticity transport, dispersal in porous media, and hydrodynamic turbulence. The analysis of these methods is implemented on the BK, Burgers systems and solutions have been offered in a simple formula. We show our results in figures and tables to demonstrate the efficiency and reliability of the used methods. Furthermore, our outcome converges rapidly to the given exact solutions.

A novel analytical Aboodh residual power series method for solving linear and nonlinear time-fractional partial differential equations with variable coefficients

<abstract><p>The goal of this research is to develop a novel analytic technique for obtaining the approximate and exact solutions of the Caputo time-fractional partial differential equations (PDEs) with variable coefficients. We call this technique as the Aboodh residual power series method (ARPSM), because it apply the Aboodh transform along with the residual power series method (RPSM). It 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 the computation of the fractional derivatives each time. As ARPSM just requires the idea of an infinite limit, we simply need a few computations to get the coefficients. This technique solves nonlinear problems without the He's polynomials and Adomian polynomials, so the small size of computation of this technique is the strength of the scheme, which is an advantage over the homotopy perturbation method and the Adomian decomposition method. The absolute and relative errors of five linear and non-linear problems are numerically examined to determine the efficacy and accuracy of ARPSM for time-fractional PDEs with variable coefficients. In addition, numerical results are also compared with other methods such as the RPSM and the natural transform decomposition method (NTDM). Some graphs are also plotted for various values of fractional orders. The results show that our technique is easy to use, accurate, and effective. Mathematica software is used to calculate the numerical and symbolic quantities in the paper.</p></abstract>

Numerical solution of one- and two-dimensional time-fractional Burgers equation via Lucas polynomials coupled with Finite difference method

In this article, a numerical technique based on polynomials is proposed for the solution of one and two-dimensional time-fractional Burgers equation. First, the given problem is reduced to time discrete form using θ-weighted scheme. Then, with the help of Lucas and Fibonacci polynomials the given PDEs transformed to system of algebraic equations which is easy to solve. The proposed algorithm is validated by solving some numerical examples. Despite this, convergence analysis of the scheme is briefly discussed and verified numerically. The main objective of this paper is to show that polynomial based method is convenient for 1D and 2D nonlinear time-fractional partial differential equations (TFPDEs). Efficiency and performance of the proposed technique are examined by calculating L2 and L∞ error norms. Obtained accurate results confirm applicability and efficiency of the method.

Semianalytical Solutions of Some Nonlinear-Time Fractional Models Using Variational Iteration Laplace Transform Method

In this work, we combined two techniques, the variational iteration technique and the Laplace transform method, in order to solve some nonlinear-time fractional partial differential equations. Although the exact solutions may exist, we introduced the technique VITM that approximates the solutions that are difficult to find. Even a single iteration best approximates the exact solutions. The fractional derivatives being used are in the Caputo-Fabrizio sense. The reliability and efficiency of this newly introduced method is discussed in details from its numerical results and their graphical approximations. Moreover, possible consequences of these results as an application of fixed-point theorem are placed before the experts as an open problem.

A new approximate analytical method and its convergence for nonlinear time-fractional partial differential equations

The main goal of this paper is to present a new approximate analytical method called modified generalized Taylor fractional series method (MGTFSM) for solving general nonlinear time- fractionalpartial differential equations. The fractional derivative is considered in the Caputo sense. We establish the convergence results of the proposed method. The basic idea of the MGTFSM is to construct the solution in the form of infinite series which converges rapidly to the exact solution of the given problem. The main advantage of the proposed method compare to current methods is that method solves the nonlinear problems without using linearization, discretization, perturbation or anyother restriction. The efficiency and accuracy of the MGTFSM is tested by means of different numerical examples. The results prove that the proposed method is very effective and simple for solving the nonlinear fractional problems.

Comparative study of two techniques on some nonlinear problems based ussing conformable derivative

AbstractIn this paper, three eminent types of time-fractional nonlinear partial differential equations are considered, which are the fractional foam drainage equation, fractional Gardner equation, and fractional Fornberg–Whitham equation in the sense of conformable fractional derivative. The approximate solutions of these considered problems are constructed and discussed using the conformable fractional variational iteration method and conformable fractional reduced differential transform method. The conformable derivative is one of the admirable choices to handle nonlinear physical problems of different fields of interest. Comparisons of approximate solution obtained by two techniques, to each other and with the exact solutions are also presented and affirm that the considered methods are efficient and reliable techniques to study other nonlinear fractional equations and models in the sense of conformable derivative. To explain the effects of several parameters and variables on the movement, the approximate results are shown in tables and two-and three-dimensional surface graphs.

A numerical method using Laplace-like transform and variational theory for solving time-fractional nonlinear partial differential equations with proportional delay

Time-fractional nonlinear partial differential equations (TFNPDEs) with proportional delay are commonly used for modeling real-world phenomena like earthquake, volcanic eruption, and brain tumor dynamics. These problems are quite challenging, and the transcendental nature of the delay makes them even more difficult. Hence, the development of efficient numerical methods is open for research. In this paper, we use the concepts of Laplace-like transform and variational theory to develop a new numerical method for solving TFNPDEs with proportional delay. The stability and convergence of the method are analyzed in the Banach sense. The efficiency of the proposed method is demonstrated by solving some test problems. The numerical results show that the proposed method performs much better than some recently developed methods and enables us to obtain more accurate solutions.

New Perspective on the Conventional Solutions of the Nonlinear Time-Fractional Partial Differential Equations

The role of integer and noninteger order partial differential equations (PDE) is essential in applied sciences and engineering. Exact solutions of these equations are sometimes difficult to find. Therefore, it takes time to develop some numerical techniques to find accurate numerical solutions of these types of differential equations. This work aims to present a novel approach termed as fractional iteration algorithm-I for finding the numerical solution of nonlinear noninteger order partial differential equations. The proposed approach is developed and tested on nonlinear fractional-order Fornberg–Whitham equation and employed without using any transformation, Adomian polynomials, small perturbation, discretization, or linearization. The fractional derivatives are taken in the Caputo sense. To assess the efficiency and precision of the suggested method, the tabulated numerical results are compared with the standard variational iteration method and the exact solution as well. In addition, numerical results for different cases of the fractional-order α are presented graphically, which show the effectiveness of the proposed procedure and revealed that the proposed scheme is very effective, suitable for fractional PDEs, and may be viewed as a generalization of the existing methods for solving integer and noninteger order differential equations.