Nonlinear Time-fractional Differential Equations Research Articles

A novel Adomian natural decomposition method with convergence analysis of nonlinear time-fractional differential equations

ABSTRACT The objective of the current article is to use the Banach fixed point theorem to present proofs for the existence and uniqueness theorems for a nonlinear time-fractional differential equation, namely, the linear and nonlinear, homogeneous and nonhomogeneous time-fractional diffusion equations. In addition, we explore exact solutions to nonlinear time fractional partial differential equations using an effective method called the Adomian natural decomposition method (ANDM), which is a combination of the Adomian decomposition method (ADM) and the natural transform method (NTM). To demonstrate the effectiveness of the suggested scheme, three examples are provided along with graphs and numerical tables to support our work. The outcomes demonstrate how effortless and effective the ANDM is for addressing fractional partial differential equations in multi-dimensional spaces, some of which we examined in this study as special cases. AMS Classifications 34A08, 65P99, 49J15, 35R11, 26A33, 74G10.

Wave solutions in (3 + 1)-dimensional generalized fractional mKdV-ZK equation utilizing jacobi elliptic functions

In this research, we investigate the effects of fractional order on the (3 + 1)-dimensional generalized space-time fractional modified KdV-Zakharov-Kuznetsov (mKdV-ZK) equation. We approach the problem by utilizing the conformable fractional derivative. By reducing the mKdV-ZK equation to an integer order nonlinear ordinary differential equation, we apply the Jacobi elliptic function method to find exact solutions. These solutions are specifically tailored for the fractional order of the (3 + 1)-dimensional generalized mKdV-ZK equation, encompassing solitary waves, shock waves, and periodic waves. We also compare these exact solutions with fractional solutions to gain further insights. Notably, our approach demonstrates the feasibility of solving nonlinear time-fractional differential equations with conformable derivatives. Several diagrams have been included to visually depict the behavior of the solutions under fractional order when certain special parameter values are employed.

A Hybrid Non-Polynomial Spline Method and Conformable Fractional Continuity Equation

This paper presents a groundbreaking numerical technique for solving nonlinear time fractional differential equations, combining the conformable continuity equation (CCE) with the Non-Polynomial Spline (NPS) interpolation to address complex mathematical challenges. By employing conformable descriptions of fractional derivatives within the CCE framework, our method ensures enhanced accuracy and robustness when dealing with fractional order equations. To validate our approach’s applicability and effectiveness, we conduct a comprehensive set of numerical examples and assess stability using the Fourier method. The proposed technique demonstrates unconditional stability within specific parameter ranges, ensuring reliable performance across diverse scenarios. The convergence order analysis reveals its efficiency in handling complex mathematical models. Graphical comparisons with analytical solutions substantiate the accuracy and efficacy of our approach, establishing it as a powerful tool for solving nonlinear time-fractional differential equations. We further demonstrate its broad applicability by testing it on the Burgers–Fisher equations and comparing it with existing approaches, highlighting its superiority in biology, ecology, physics, and other fields. Moreover, meticulous evaluations of accuracy and efficiency using (L2 and L∞) norm errors reinforce its robustness and suitability for real-world applications. In conclusion, this paper presents a novel numerical technique for nonlinear time fractional differential equations, with the CCE and NPS methods’ unique combination driving its effectiveness and broad applicability in computational mathematics, scientific research, and engineering endeavors.

Conformable non-polynomial spline method: A robust and accurate numerical technique

The article introduces a novel and developed a numerical technique that solves nonlinear time fractional differential equations. It combines finite difference and non-polynomial spline methods while using conformable descriptions of fractional derivatives. The strategy has proven to be useful to analyzing intricate datasets across various domains, including finance, science, and engineering. The applicability and validity of the approach are demonstrated through numerical examples. The study assessed the approach's stability using the Fourier method and found it to be unconditionally stable for a specific parameter range. The proposed method has been analyzed for convergence, and the analysis reveals that it exhibits a convergence order of six. The paper also includes graphs comparing the present solution to an analytical one. The method is tested with some examples of the model that have a wide range of applications in fields such as biology, ecology, and physics called the Burgers-Fisher equations and compared to previous approaches to demonstrate its effectiveness and applicability. The research evaluated the approach's accuracy and efficiency using the (L2 and L∞) norm errors. Also, the effects of time and fractional derivatives have been studied, and to the best of our knowledge, it has not been previously published in any academic or industry publication.

A numerical solution of fractional reaction–convection–diffusion for modeling PEM fuel cells based on a meshless approach

The purpose of this contribution is to present or implement generalized finite difference method (GFDM) for the first time in order to solve the reaction convection Diffusion equation (FRCDE) model. The FRCDE equations are nonlinear time fractional differential equations that can be used to describe mathematical models in physics and engineering, such as fuel cells and transport in inhomogeneous media. This model is a generalization of the classical RCDE, with time terms considered by Caputo derivative sense for 0<α(x,t)<1. The meshless GFDM, in conjunction with the ϑ-weighted finite difference method, is developed to approximate processes in the spatial direction. GFDM is based on the Taylor series expansion and the moving least squares (MLS) method to evaluate the derivatives of unknown variables. The proposed method, as a truly meshless approach, is very promising in the numerical approximation of engineering problems within convex, non-convex, irregular, and regular domains (complex domains). The reliability and accuracy of the proposed method are shown by considering a variety of computational domains. The sensitivity of the selection of local collocation points and ϑ is also reported.

Solutions of Time Fractional fKdV Equation Using the Residual Power Series Method

The fifth-order Korteweg-de Vries (fKdV) equation is a nonlinear model in various long wave physical phenomena. The residual power series method (RPSM) is used to gain the approximate solutions of the time fractional fKdV equation in this study. Basic definitions of fractional derivatives are described in the Caputo sense. The solutions of the time fractional fKdV equation with easily computable components are calculated as a quick convergent series. When compared to exact solutions, the RPSM provides good accuracy for approximate solutions. The reliability of the proposed method is also illustrated with the aid of table and graphs. Cleary observed from the results that the suggested method is suitable and simple for similar type of the time fractional nonlinear differential equations.

The Solution of Nonlinear Time-Fractional Differential Equations: An Approximate Analytical Approach

The Solution of Nonlinear Time-Fractional Differential Equations: An Approximate Analytical Approach

The Numerical Solutions of Nonlinear Time-Fractional Differential Equations by LMADM

This paper presents a numerical scheme for solving nonlinear time-fractional differential equations in the sense of Caputo. This method relies on the Laplace transform together with the modified Adomian method (LMADM), compared with the Laplace transform combined with the standard Adomian Method (LADM). Furthermore, for the comparison purpose, we applied LMADM and LADM for solving nonlinear time-fractional differential equations to identify the differences and similarities. Finally, we provided two examples regarding the nonlinear time-fractional differential equations, which showed that the convergence of the current scheme results in high accuracy and small frequency to solve this type of equations.

Exact solutions of the generalized multidimensional mathematical physics models via sub-equation method

In this paper, our focus is on the multidimensional mathematical physics models. We employ the sub-equation method to obtain new exact solutions to the proposed strongly nonlinear time-fractional differential equations of conformable type. The models considered are generalized Benjamin equation, modified generalized multidimensional Kadomtsev–Petviashvili (KP) equations, modified generalized multidimensional KP–BBM equation and the variant Boussinesq system of equations. We also introduced a new modified generalized multidimensional KP type equation and its exact solutions. As the order of fractional derivative tends to one, the obtain exact solutions by the proposed method reduce to classical solutions. We successfully established varieties of soliton type solutions. The results obtained affirm that sub-equation method is an efficient and powerful technique for analytic solutions of nonlinear fractional partial differential equations.

An Efficient Numerical Technique for Solving Time-Fractional Generalized Fisher's Equation

This paper extends the existing Fisher equation by adding the source term and generalizing the degree $\beta$ of non-linear part. Numerical solutions of modified Fisher's equation for different values of $\beta$ using cubic B-spline collocation scheme are investigated. The fractional derivative in time dimension is discretized in Caputo's form based on $L1$ formula, while cubic B-spline basis functions are used to interpolate the spatial derivative. The non-linear part in the equation is linearized by the modified formula. The efficiency of the proposed scheme is examined by considering four test examples with different initial and boundary conditions. The effect of different parameters is discussed and presented in the form of tables and graphics. Moreover, by Von Neumann stability formula, proposed scheme is shown to be unconditionally stable. The results of error norms reflect that present scheme is suitable for non-linear time fractional differential equations.

A novel approach for the analytical solution of nonlinear time-fractional differential equations

PurposeThe purpose of this paper is to suggest the solution of time-fractional Fornberg–Whitham and time-fractional Fokker–Planck equations by using a novel approach.Design/methodology/approachFirst, some basic properties of fractional derivatives are defined to construct a novel approach. Second, modified Laplace homotopy perturbation method (HPM) is constructed which yields to a direct approach. Third, two numerical examples are presented to show the accuracy of this derived method and graphically results showed that this method is very effective. Finally, convergence of HPM is proved strictly with detail.FindingsIt is not necessary to consider any type of assumptions and hypothesis for the development of this approach. Thus, the suggested method becomes very simple and a better approach for the solution of time-fractional differential equations.Originality/valueAlthough many analytical methods for the solution of fractional partial differential equations are presented in the literature. This novel approach demonstrates that the proposed approach can be applied directly without any kind of assumptions.

Optical soliton solutions of perturbing time-fractional nonlinear Schrödinger equations

This article presents a technique for finding solitary wave solutions of higher-order nonlinear differential equations. Time fractional nonlinear differential equations that have hierarchy contain higher orders are presented for the description of highly dispersive optical solutions. We find optical solitons with high dispersion order for time-fractional nonlinear perturbed Schrödinger equations of the fourth and sixth orders. Some figures are investigated to show the accuracy of the solutions obtained.

Exact Traveling Wave Solutions and Bifurcations of the Time-Fractional Differential Equations with Applications

This paper presents a method to investigate exact traveling wave solutions and bifurcations of the nonlinear time-fractional partial differential equations with the conformable fractional derivative proposed by [Khalil et al., 2014]. The method is based on employing the bifurcation theory of planar dynamical systems proposed by [Li, 2013]. For the fractional PDEs, up till now, there is no related paper to obtain the exact solutions by applying bifurcation theory. We show how to use this method with applications to two fractional PDEs: the fractional Klein–Gordon equation and the fractional generalized Hirota–Satsuma coupled KdV system, respectively. We find the new exact solutions including periodic wave solution, kink wave solution, anti-kink wave solution and solitary wave solution (bright and dark), which are different from previous works in the literature. This approach can also be extended to other nonlinear time-fractional differential equations with the conformable fractional derivative.

New wave form solutions of nonlinear conformable time-fractional Zoomeron equation in (2 + 1)-dimensions

Under investigation in the current paper is the nonlinear conformable time-fractional Zoomeron equation in (2 + 1)-dimensions, which is a model to display the novel phenomena associated with boomerons and trappons. The well-designed techniques, -expansion approach and modified Kudryashov method are formally utilized to produce a variety of wave form solutions such as kink, singular kink, and periodic wave solutions for the governing model. Results confirm the effectiveness of the methods for extracting different wave form solutions of nonlinear time-fractional differential equations.

Application of residual power series method to time fractional gas dynamics equations

In this study, we proposed the power series method called Residual Power Series(RPS) method coupled with Fractional Complex Transform(FCT) to solve the time fractional nonlinear gas dynamics equations. FCT is the alternative approach in fractional calculus which transforms the fractional differential equation in to integer order differential equation making the solution in simple manner. The results reveal that RPS method is simple and convenient for similar type of time fractional nonlinear differential equations.

Solitary and periodic wave solutions of higher-dimensional conformable time-fractional differential equations using the ( frac{G'}{G},frac{1}{G} ) -expansion method

In this paper, the two variables ( frac{G'}{G},frac{1}{G} ) -expansion method is applied to obtain new exact solutions with parameters of higher-dimensional nonlinear time-fractional differential equations (NTFDEs) in the sense of the conformable fractional derivative. To clarify the veracity of this method, it is implemented in nonlinear (2+1)-dimensional time-fractional biological population (BP) model and nonlinear (3+1)-dimensional KdV–Zakharov–Kuznetsov (KdV–ZK) equation with time-fractional derivative. When the parameters take some special values, the solitary and periodic solutions are obtained from the hyperbolic and trigonometric function solutions.

Parallel algorithms for nonlinear time–space fractional parabolic PDEs

In this paper, we develop a time stepping scheme for solving nonlinear time–space fractional partial differential equations (PDEs). In space, we use the matrix transfer technique to discretize the PDEs and obtain a system of nonlinear time-fractional differential equations. The developed scheme is similar to the Crank–Nicholson scheme for integer order PDEs and are shown to be of order 1+α in time where α is the order of the time derivative described in the Caputo sense. The solution of the PDE at any point i in the t-stencil depends not only on the solution at the point i−1 but on all previous solutions (memory effect). Thus, the implementation of schemes for such fractional PDEs, for long time interval, can be time consuming. This is basically due to the computation and re-computation of the history term at each time step. We lessen this computational time by implementing three parallel versions of the algorithm. The shared memory systems (OpenMP) and the distributed memory systems (MPI) are used for implementing the parallel algorithms. A third parallel version uses both the shared and distributed memory systems (Hybrid version). The advantages of the parallel algorithms over the sequential algorithm are discussed. The merits and demerits of each parallel versions of the algorithm over the others are examined. Numerical simulations are performed to support our theoretical observations.

Simplest equation method for some time-fractional partial differential equations with conformable derivative

The conformable fractional derivative was proposed by R. Khalil et al. in 2014, which is natural and obeys the Leibniz rule and chain rule. Based on the properties, a class of time-fractional partial differential equations can be reduced into ODEs using traveling wave transformation. Then the simplest equation method is applied to find exact solutions of some time-fractional partial differential equations. The exact solutions (solitary wave solutions, periodic function solutions, rational function solutions) of time-fractional generalized Burgers equation, time-fractional generalized KdV equation, time-fractional generalized Sharma–Tasso–Olver (FSTO) equation and time-fractional fifth-order KdV equation, (3+1)-dimensional time-fractional KdV–Zakharov–Kuznetsov (KdV–ZK) equation are constructed. This method presents a wide applicability to solve some nonlinear time-fractional differential equations with conformable derivative.

A Generalized Definition of Caputo Derivatives and Its Application to Fractional ODEs

We propose a generalized definition of Caputo derivatives from $t=0$ of order $\gamma\in(0,1)$ using a convolution group, and we build a convenient framework for studying initial value problems of general nonlinear time fractional differential equations. Our strategy is to define a modified Riemann--Liouville fractional calculus which agrees with the traditional Riemann--Liouville definition for $t>0$ but includes some singularities at $t=0$ so that the group property holds. Then, making use of this fractional calculus, we introduce the generalized definition of Caputo derivatives. The new definition is consistent with various definitions in the literature while revealing the underlying group structure. The underlying group property makes many properties of Caputo derivatives natural. In particular, it allows us to deconvolve the fractional differential equations to integral equations with completely monotone kernels, which then enables us to prove the general comparison principle with the most general conditions. This then allows for a priori energy estimates of fractional PDEs. Since the new definition is valid for locally integrable functions that can blow up in finite time, it provides a framework for solutions to fractional ODEs and fractional PDEs. Many fundamental results for fractional ODEs are revisited within this framework under very weak conditions.

Exact wave solutions for the nonlinear time fractional Sharma–Tasso–Olver equation and the fractional Klein–Gordon equation in mathematical physics

In this study, we examine the resourceful traveling wave solutions of a pair of nonlinear time fractional differential equations, namely, the general time fractional Sharma–Tasso–Olver equation and the fractional Klein–Gordon equation in the form of hyperbolic functions, trigonometric functions, and rational functions along with free parameters. To establish above mentioned type of solutions those are factual, convenient to use and penetrate the fission and fusion phenomena take place in solitons, the relativistic energy–momentum relation, scalar electrodynamics, electromagnetic interactions, plasma physics, nuclear physics etc. very clearly, we implement the new generalized $$\left( {G'/G} \right)$$ -expansion method. We emphasize that this method is highly effective and a forcible mathematical tool to extract solitary wave solutions of the time fractional models that are extensively used in theoretical physics.