A note on exp-function method combined with complex transform method applied to fractional differential equations

The idea proposed in this work is to extend the ZZ transform method to resolve the nonlinear fractional partial differential equations by combining them with the so-called homotopy perturbation method (HPM). We apply this technique to solve some nonlinear fractional equations as: nonlinear time-fractional Fokker-Planck equation, the cubic nonlinear time-fractional Schrodinger equation and the nonlinear timefractional KdV equation. The fractional derivative is described in the Caputo sense. The results show that this is the appropriate method to solve somme models of nonlinear partial differential equations with time-fractional derivative.

PurposeThe purpose of this paper is the comparative analysis of Haar Wavelet Method and Optimal Homotopy Asymptotic Method for fractional Fisher type equation. In this paper, two reliable techniques, Haar wavelet method and optimal homotopy asymptotic method (OHAM), have been presented. The Haar wavelet method is an efficient numerical method for the numerical solution of fractional order partial differential equation like the Fisher type. The approximate solutions of the fractional Fisher-type equation are compared with those of OHAM and with the exact solutions. Comparisons between the obtained solutions with the exact solutions exhibit that both the featured methods are effective and efficient in solving nonlinear problems. However, the results indicate that OHAM provides more accurate value than the Haar wavelet method.Design/methodology/approachComparisons between the solutions obtained by the Haar wavelet method and OHAM with the exact solutions exhibit that both featured methods are effective and efficient in solving nonlinear problems.FindingsThe comparative results indicate that OHAM provides a more accurate value than the Haar wavelet method.Originality/valueIn this paper, two reliable techniques, the Haar wavelet method and OHAM, have been proposed for solving nonlinear fractional partial differential equation, i.e. fractional Fisher-type equation. The proposed novel methods are well suited for only nonlinear fractional partial differential equations. It also exhibits that the proposed method is a very efficient and powerful technique in finding the solutions for the nonlinear time fractional differential equations. The main significance of the proposed method is that it requires less amount of computational overhead in comparison to other numerical and analytical approximate methods. The application of the proposed methods for the solutions of time fractional Fisher-type equations satisfactorily justifies its simplicity and efficiency.

Recent advancement in the field of nonlinear analysis and fractional calculus help to address the rising challenges in the solution of nonlinear fractional partial differential equations. This paper presents a hybrid technique, a combination of Tarig transform and Projected Differential Transform Method (TPDTM) to solve nonlinear fractional partial differential equations. The effectiveness of the method is examined by solving three numerical examples that arise in the field of heat transfer analysis. In this proposed scheme, the solution is obtained as a convergent series and the result is used to analyze the hyper diffusive process with pre local information regarding the heat transfer for different values of fractional order. In order to validate the results, a comparative study has been carried out with the solution obtained from the two methods, the Laplace Adomian Decomposition Method (LADM) and Homotophy Pertubation Method (HPM) and the result thus observed coincided with each other. Inspite of the uniformity between the solutions, the proposed hybrid technique had to overcome the complexity of manupulation of Adomian polynomials and evaluation of integrals in LADM and HPM respectively. The methodology and the results presented in this paper clearly reveals the computational efficiency of the present method. Due to its computational efficiency, the TPDTM has the potential to be used as a novel tool not only for solving nonlinear fractional differential equations but also for analysing the prelocal information of the system.

Nonlinear fractional differential equations are widely used for modelling problems in applied mathematics. A new analytic approach with two parameters c1 and c2 is first proposed for solving nonlinear fractional partial differential equations. These parameters are used to improve the accuracy of the resulting series approximations. It turns out that much more accurate series approximations are obtained by choosing proper values of c1 and c2. To demonstrate the applicability and effectiveness of the new method, two typical fractional partial differential equations, the nonlinear gas dynamics equation and the nonlinear KdV-Burgers equation, are solved.

Analytical solutions of the space–time fractional Kundu–Eckhaus equation by using modified extended direct algebraic method

In a range of nonlinear fields, for example molecular biology, physics in plasma, quantum mechanics, elastic media, nonlinear optics, the surface of water waves, and others, many complicated nonlinear behaviors can be pronounced using nonlinear fractional partial differential equations. The nonlinear traveling waves, including nonlinear space-time fractional order differential equations such as the time fractional Sharma-Tasso-Olver (STO) and the space-time fractional Kowerteg-de Vries-Burgers (KdV-Burgers) equations, are analysis through a resourceful approach, namely the extended tanh-function by the definition of the conformable derivative. The stated equations are prevalently used in marine and coastal infrastructure, such as fluid flow, plasma engineering, fiber optics, fusion and fission phenomena, and so on. By renovating fractional order differential equations to ordinary differential equations, fractional order differential transformation simplifies the solution process. Several types of solutions, such as soliton types, bell types, kink types, and some other kinds of results, are developed and portrayed for definite values of free parameters that are plotted in terms of 3D, contour, and spherical shapes. The method is an algebraic approach that is both direct and efficient for dealing with nonlinear equations and yield a distinct category of solutions known as solitons. The suggested method is being used in this study to generate attractive, and effective results that are flexible, easy, and quicker to simulate using general mathematical techniques, namely Maple. It is worth declaring that all resultant solutions are verified for accuracy by exchanging them directly within the original equation through the software package and originating them accurately.MSC: 35C25, 35C07, 35C08, 35Q20, 76B25

A table lookup method for solving nonlinear fractional partial differential equations (fPDEs) is proposed in this paper. Looking up the corresponding tables, we can quickly obtain the exact analytical solutions of fPDEs by using this method. To illustrate the validity of the method, we apply it to construct the exact analytical solutions of four nonlinear fPDEs, namely, the time fractional simplified MCH equation, the space-time fractional combined KdV-mKdV equation, the (2+1)-dimensional time fractional Zoomeron equation, and the space-time fractional ZKBBM equation. As a result, many new types of exact analytical solutions are obtained including triangular periodic solution, hyperbolic function solution, singular solution, multiple solitary wave solution, and Jacobi elliptic function solution.

This study uses the optimal auxiliary function method to approximate solutions for fractional-order non-linear partial differential equations, utilizing Riemann–Liouville’s fractional integral and the Caputo derivative. This approach eliminates the need for assumptions about parameter magnitudes, offering a significant advantage. We validate our approach using the time-fractional Cahn–Hilliard, fractional Burgers–Poisson, and Benjamin–Bona–Mahony–Burger equations. Comparative testing shows that our method outperforms new iterative, homotopy perturbation, homotopy analysis, and residual power series methods. These examples highlight our method’s effectiveness in obtaining precise solutions for non-linear fractional differential equations, showcasing its superiority in accuracy and consistency. We underscore its potential for revealing elusive exact solutions by demonstrating success across various examples. Our methodology advances fractional differential equation research and equips practitioners with a tool for solving non-linear equations. A key feature is its ability to avoid parameter assumptions, enhancing its applicability to a broader range of problems and expanding the scope of problems addressable using fractional calculus techniques.

The space–time fractional Telegraph equation and the space–time fractional modified third-order Kdv equations are significant molding equations in theoretic physics, mathematical physics, plasma physics also other fields of nonlinear sciences. The space time-fractional telegraph equation, which appears in the investigation of an electrical communication line and includes voltage in addition to current which is dependent on distance and time, is also applied to communication lines of wholly frequencies, together with direct current, as well as high-frequency conductors, audio frequency (such as telephone lines), and low frequency (for example cable television) used in the extension of pressure waves into the lessons of pulsatory blood movement among arteries also the one-dimensional haphazard movement of bugs towards an obstacle. The presence of chain rule and the derivative of composite functions allows the nonlinear fractional differential equations to translate into the ordinary differential equation employing wave alteration. To explore such categories of resolutions, the extended tanh-method is accomplished via Conformable derivatives. A power sequence in tanh was originally used as an ansatz to provide analytical solutions of the traveling wave type of certain nonlinear evolution equations. The outcomes achieved in this study are king type, single soliton, double soliton, multiple solitons, bell shape, and other sorts of forms and we demonstrated that these solutions were validated through the Maple software. The proposed approach for solving nonlinear fractional partial differential equations has been developed to be operative, unpretentious, quick, and reliable to usage.

Given a time fractional nonlinear partial differential equation, we show how to derive its exact solution using invariant subspace method. This has been illustrated through time fractional diffusion convection equation, time fractional nonlinear dispersive Boussinesq equation, time fractional reaction diffusion equation of second order, time fractional thin-film equation, and time fractional quadratic wave equation. Also, we explicitly shown that time fractional nonlinear partial differential equations admit more than one invariant subspaces which in turn helps to derive more than one exact solution.

A generalized local fractional Riccati differential equation (LFRDE) method, which is a combination of the local fractional Riccati differential equation method and the transformed rational function approach, is presented to obtain the non-differentiable exact traveling wave solutions for n-dimensional fractional partial differential equations. In order to verify the effectiveness of the method, the seventh-order local fractional Sawada–Kotera–Ito equation, the local fractional sine-Gordon equation and the local fractional Kadomtsev–Petviashvili equation in fractal domain are first considered. The obtained results show that the presented method involving fractal special functions, are powerful and effective for obtaining exact solutions of nonlinear fractional partial differential equations in fractal domains.

In this article, we consider the exact solutions of the Hunter-Saxton and Schrödinger equations defined by Atangana's comformable derivative using the general Kudryashov method. Firstly, Atangana's comformable fractional derivative and its properties are included. Then, by introducing the generalized Kudryashov method, exact solutions of nonlinear fractional partial differential equations (FPDEs), which can be expressed with the comformable derivative of Atangana, are classified. Looking at the results obtained, it is understood that the generalized Kudryashov method can yield important results in obtaining the exact solutions of FPDEs containing beta-derivatives.

Variable separation method for a nonlinear time fractional partial differential equation with forcing term

In the present paper, we established a traveling wave solution by using modified Kudryashov method for the space-time fractional nonlinear partial differential equations. The method is used to obtain the exact solutions for different types of the space-time fractional nonlinear partial differential equations such as, the space-time fractional coupled equal width wave equation (CEWE) and the space-time fractional coupled modified equal width wave equation (CMEW), which are the important soliton equations. Both equations are reduced to ordinary differential equations by the use of fractional complex transform and properties of modified Riemann–Liouville derivative. We plot the exact solutions for these equations at different time levels.

The primary objective of this paper is to develop an efficient numerical method for solving two-dimensional nonlinear fractional partial differential equations in both temporal and spatial domains. The proposed approach combines a fast algorithm with Shifted Legendre wavelets to efficiently handle fractional derivatives in space and time. The resulting scheme is straightforward to implement and computationally efficient. An interpolation technique is employed to convert the nonlinear problems into equivalent linear forms, after which the proposed method is applied. Convergence analysis confirms that the numerical solutions are in good agreement with the corresponding analytical solutions. Numerical results, presented in both tabular and graphical formats, further demonstrate the accuracy and efficiency of the proposed method. In addition, comparisons with existing methods from the literature are provided to highlight the superior performance of the proposed approach.