Fractional Partial Differential Equations Research Articles

Novel approximate solution for fractional differential equations by the optimal variational iteration method

In this work, the optimal variational iteration method (OVIM) is used to solve partial and ordinary fractional differential equations. The standard variational iteration method (VIM) is reassessed by introducing a parameter that accelerates convergence. The value of the convergence acceleration coefficient is determined by calculating the residual of the parameter within the L2 norm. The numerical results proved that the proposed method gives much better and more accurate results than the standard VIM method, as the effect of adding the parameter to the VIM method and its calculation method was clear in improving the method and increasing the convergence of the solution, where we solved several examples of ordinary and partial fractional differential equations. In addition, the fractional Kawahara and foam drainage equations have also been solved. All results were obtained using the Mathematica®12 program. The results were also compared with the optimal Adomian decomposition method (ADM), and the proposed method showed higher efficiency and better accuracy.

Applications on Double ARA–Sumudu Transform in Solving Fractional Partial Differential Equations

In this article, we apply the double ARA–Sumudu transformation (DARA-ST) to the nonlocal fractional Caputo derivative operator. We achieve interesting results and implement them to solve certain classes of fractional partial differential equations (FPDEs). Several physical applications are discussed and analyzed, such as telegraph, Klein–Gordon and Fokker–Planck equations. The new technique with DARA-ST is efficient and accurate in examining exact solutions of FPDEs. In order to show the applicability of the presented method, some numerical examples and figures are illustrated. A symmetry analysis is used to verify the results.

Double-Preconditioning Techniques for Fractional Partial Differential Equation Solvers

This paper is devoted to the numerical computation of the solution to stationary space fractional Partial Differential Equations (PDEs). To this end, we derive some algorithms for solving fractional algebraic linear systems resulting from the discretization of these space fractional PDEs. The proposed approach is based on an efficient computation of Cauchy’s integrals allowing to estimate positive real powers of a (sparse) matrix A. A first preconditioner M is used to reduce the length of the Cauchy integral contour enclosing the spectrum of MA, hence allowing in some cases for a large reduction of the number of quadrature nodes along the integral contour. Next, ILU-factorizations are used to efficiently solve the linear systems involved in the computation of approximate Cauchy’s integrals. Therefore, the strategy is independent of the way the Cauchy integrals are computed, although in this paper, for the sake of simplicity, we use standard quadrature rules. Based on this approach, we derive some solvers for stationary deterministic or stochastic space fractional Poisson equations which are tested in a series of numerical experiments.

Solving Two-Sided Fractional Super-Diffusive Partial Differential Equations with Variable Coefficients in a Class of New Reproducing Kernel Spaces

Fractional-order calculus has become a useful mathematical framework to describe the complex super-diffusive process; however, numerical solutions of the two-sided space-fractional super-diffusive model with variable coefficients are difficult to obtain, and almost no method can obtain an analytical solution. In this paper, a class of new fractional dimensional reproducing kernel spaces (RKS) based on Caputo fractional derivatives is given, and we give analytical and numerical solutions of the two-sided space-fractional super-diffusive model based on the class of new RKS. The analytical solution is represented in the form of series in the reproducing kernel space. Numerical experiments indicate that the piecewise reproducing kernel method is more accurate than the traditional reproducing kernel method (RKM), and these new fractional reproducing kernel spaces are efficient for the two-sided space-fractional super-diffusive model.

Analysis of local fractional Klein-Gordon equations arising in relativistic fractal quantum mechanics

In this paper, we present the implementation of a local fractional homotopy perturbation method pertaining to the local fractional natural transform (LFNT) operator for local fractional Klein-Gordon equations (LFKGEs) under distinct fractal initial conditions. The Klein-Gordon equation is a relativistic wave equation which estimates the nature and movement of zero spin particles at high velocities and energies resembling the lightspeed. This equation admits the laws of special relativity and describes zero spin particles with relativistic energy. This work also examines uniqueness and convergence analyses of the obtained solution for a general local fractional partial differential equation via applied method. The numerical simulations are presented for each LFKG equation on the Cantor set. The computational process ensures that the copulation of local fractional homotopy perturbation method and LFNT operator provides series solutions in closed form with fast convergence for local fractional Klein-Gordon equation in a very lucid way. Furthermore, the results have also been compared with solutions obtained by other methods.

Casson Fluid Convective Flow in an Accelerated Microchannel with Thermal Radiation using the Caputo Fractional Derivative

The effect of the Caputo fractional derivative in unsteady boundary layer Casson fluid flow in an accelerated microchannel is investigated. In the presence of thermal radiation, the partial differential equations that governed the problem are studied. Using appropriate dimensionless variables, fractional partial differential equations are translated into dimensionless governing equations. The equations are then transformed into linear ordinary differential equations and solved analytically using the Laplace transform technique. These modified equations are then solved using the proper method, and the result is obtained in the form of velocity and temperature profiles using the Zakian’s explicit formula approach. The influence of essential physical parameters on velocity and temperature profiles is investigated using graphical diagrams created with Mathcad software. It is found that the velocity and temperature profile increase as fractional parameter, and thermal radiation parameter increase. As Prandtl number increase, both profiles are decreasing. This result is crucial for understanding the fractional system of Casson fluid in microchannel.

Time-Fractional Differential Equations with an Approximate Solution

This paper shows how to use the fractional Sumudu homotopy perturbation technique (SHP) with the Caputo fractional operator (CF) to solve time fractional linear and nonlinear partial differential equations. The Sumudu transform (ST) and the homotopy perturbation technique (HP) are combined in this approach. In the Caputo definition, the fractional derivative is defined. In general, the method is straightforward to execute and yields good results. There are some examples offered to demonstrate the technique's validity and use.

The Dynamics of a Fractional-Order Mathematical Model of Cancer Tumor Disease

This article explores the application of the reduced differential transform method (RDTM) for the computational solutions of two fractional-order cancer tumor models in the Caputo sense: the model based on cancer chemotherapeutic effects which explain the relation between chemotherapeutic drugs, tumor cells, normal cells, and immune cells using a fractional partial differential equations, and the model that describes the different cases of killing rate K of cancer cells (the killing percentage of cancer cells K (I) is dependent on the number of cells, (II) is a function of time only, and (III) is a function of space only). The solutions are presented using Mathematica software as a convergent power series with elegantly computed terms using the suggested technique. The proposed method gives new series form results for various values of gamma. To clarify the complexity of the models, we plot the two- and three-dimensional and contour graphics of the obtained solutions at varied values of fractional-order gamma and the selected system parameters. The solutions are analyzed with fractional and reduced differential transform methods to obtain an idea of invariance regarding the computed solution of the designed mathematical model. The obtained results demonstrate the efficiency and preciseness of the proposed method to achieve a better understanding of chemotherapy effects. It is observed that chemotherapy drugs boost immunity against the specific cancer by decreasing the number of tumor cells, and the killing rate K of cancerous cells depend on the cells concentration.

Lie symmetry analysis of time fractional Burgers equation, Korteweg-de Vries equation and generalized reaction-diffusion equation with delays

In this paper, Lie symmetry analysis method is applied to time fractional Burgers equation, Korteweg-de Vries equation and generalized reaction-diffusion equation with delays, respectively. The Lie symmetries for fractional partial differential equations with delays (DFPDEs) are obtained, and the group classifications of the equations are established. The obtained group generators are used to reduce the DFPDEs to fractional ordinary differential equations with delays (DFODEs). Some exact solutions constructed for the DFODEs generate group-invariant solutions of the discussed DFPDEs.

A new modified technique to study the dynamics of fractional hyperbolic-telegraph equations

Abstract Usually, to find the analytical and numerical solution of the boundary value problems of fractional partial differential equations is not an easy task; however, the researchers devoted their sincere attempt to find the solutions of various equations by using either analytical or numerical procedures. In this article, a very accurate and prominent method is developed to find the analytical solution of hyperbolic-telegraph equations with initial and boundary conditions within the Caputo operator, which has very simple calculations. This method is called a new technique of Adomian decomposition method. The obtained results are described by plots to confirm the accuracy of the suggested technique. Plots are drawn for both fractional and integer order solutions to confirm the accuracy and validity of the proposed method. Solutions are obtained at different fractional orders to discuss the useful dynamics of the targeted problems. Moreover, the suggested technique has provided the highest accuracy with a small number of calculations. The suggested technique gives results in the form of a series of solutions with easily computable and convergent components. The method is simple and straightforward and therefore preferred for the solutions of other problems with both initial and boundary conditions.

New Efficient Computations with Symmetrical and Dynamic Analysis for Solving Higher-Order Fractional Partial Differential Equations

Due to the rapid development of theoretical and computational techniques in the recent years, the role of nonlinearity in dynamical systems has attracted increasing interest and has been intensely investigated. A study of nonlinear waves in shallow water is presented in this paper. The classic form of the Korteweg–de Vries (KdV) equation is based on oceanography theory, shallow water waves in the sea, and internal ion-acoustic waves in plasma. A shallow fluid assumption is shown in the framework by a sequence of nonlinear fractional partial differential equations. Indeed, the primary purpose of this study is to use a semi-analytical technique based on Fractional Taylor Series to achieve numerical results for nonlinear fifth-order KdV models of non-integer order. Caputo is the operator used for dealing with fractional derivatives. The generated solutions of nonlinear fifth-order KdV models of non-integer order for modeling turbulence processes in the field of ocean engineering are compared analytically and numerically, to demonstrate the behaviors of several parameters of the current model. We verified the method’s convergence analysis and provided an error estimate by showing 2D and 3D graphs to further confirm its efficacy.

The solutions of nonlinear fractional partial differential equations by using a novel technique

Abstract In this article, the solutions of higher nonlinear partial differential equations (PDEs) with the Caputo operator are presented. The fractional PDEs are modern tools to model various phenomena more accurately. The residual power series method (RPSM) is used for the solution analysis of fractional partial differential equations (FPDEs), which has direct implementation for the solutions of fractional partial differential equations. In this work, the solutions to a few nonlinear FPDEs are handled by the proposed technique. The general and particular schemes of RPSM are constructed and implemented successfully. The fractional solutions of PDEs have provided many useful dynamics of the targeted problems. The RPSM results for both integer and fractional-order FPDEs are further explained and elaborated by using graphs and tables. It is observed that the higher accuracy of RPSM is achieved with fewer calculations. Graphs and tables for fractional-order solutions are presented, which confirm the convergence phenomena of fractional solutions toward integer order solutions of each problem. The suggested method can be extended to the solutions of other nonlinear fractional partial differential equations.

THE INFLUENCE OF FRACTIONAL ORDER AND MAGNETIC FIELD ON ELECTRO HYDRODYNAMIC FLOW OF JEFFREY FLUID ALONG WITH MAGNETIC NANOPARTICLES THROUGH ARTERIAL SEGMENT

The study reports the influence of Caputo–Fabrizio fractional derivative and magnetic field on blood flow as an electrically conducting non-Newtonian fluid along with magnetic nanoparticles through circular cylindrical arterial segment, by assuming blood as Jeffrey fluid. The main structure of the governing fractional nonlinear partial differential equations was obtained from non-Newtonian fluid model in which convective derivative is considered instead of the time derivative which is capable of describing the phenomena of relaxation time and retardation time. The exact solutions of the governing fractional partial differential equations were obtained by the virtue of Laplace and Hankel transforms. Numerical simulations have been performed to analyze the behavior of the Jeffrey fluid flow using Mathcad software and the results are presented graphically for an explicit and detail discussion. We noticed from the graphical representation of results that the fractional derivative, particle concentration, electro-kinetic width, Jeffrey number and the applied magnetic field have profound influence on the magnitude of blood velocity along with magnetic nanoparticles. The effect of each of the influential parameters tremendously decelerates the fluid flow and thereby showing appreciable variation in the distribution of axial and radial velocities. Practically, increasing the applied magnetic field strength and the value of the Jeffrey number on the motion of fluid flow together with magnetic nanoparticles, vehemently decreases the velocities of blood along with magnetic nanoparticles, which helps to have effective control over the fluid networks. From the study, we conclude that the fractional model, Jeffrey number and the application of the external magnetic field on the fluid flow have extensive applications in the clinical and the medical sciences.

Parameter Estimation for Several Types of Linear Partial Differential Equations Based on Gaussian Processes

This paper mainly considers the parameter estimation problem for several types of differential equations controlled by linear operators, which may be partial differential, integro-differential and fractional order operators. Under the idea of data-driven methods, the algorithms based on Gaussian processes are constructed to solve the inverse problem, where we encode the distribution information of the data into the kernels and construct an efficient data learning machine. We then estimate the unknown parameters of the partial differential Equations (PDEs), which include high-order partial differential equations, partial integro-differential equations, fractional partial differential equations and a system of partial differential equations. Finally, several numerical tests are provided. The results of the numerical experiments prove that the data-driven methods based on Gaussian processes not only estimate the parameters of the considered PDEs with high accuracy but also approximate the latent solutions and the inhomogeneous terms of the PDEs simultaneously.

Solving Nonlinear Fractional Partial Differential Equations Using the Elzaki Transform Method and the Homotopy Perturbation Method

In this paper, we combine the Elzaki transform method (ETM) with the new homotopy perturbation method (NHPM) for the first time. This hybrid approach can solve initial value problems numerically and analytically, such as nonlinear fractional differential equations of various normal orders. The Elzaki transform method (ETM) is used to solve nonlinear fractional differential equations, and then the homotopy is applied to the transformed equation, which includes the beginning conditions. To obtain the solution to an equation, we use the inverse transforms of the Elzaki transform method (ETM). The initial conditions have a big impact on the equation’s result. We give three beginning value issues that were solved as precise or approximation solutions with high rigor to demonstrate the method’s power and correctness. It is clear that solving nonlinear partial differential equations with the crossbred approach is the best alternative.

Shehu transform on time-fractional Schrödinger equations – an analytical approach

Abstract In the present study, time-fractional Schrödinger equations are dealt with for the analytical solution using an integral transform named Shehu Transform. Three kinds of time-fractional Schrödinger equations are discussed in the present study. Shehu transform is utilized to reduce the time-fractional PDE along with the fractional derivative in the Caputo sense. The present method is easy to implement in the search for an analytical solution. As no discretization or numerical program is required, the present scheme will surely be helpful in finding the analytical solution to some complex-natured fractional PDEs.

A New Modified Technique of Adomian Decomposition Method for Fractional Diffusion Equations with Initial-Boundary Conditions

In general, solving fractional partial differential equations either numerically or analytically is a difficult task. However, mathematicians have tried their best to make the task easy and promoted various techniques for their solutions. In this regard, a very prominent and accurate technique, which is known as the new technique of the Adomian decomposition method, is developed and presented for the solution of the initial-boundary value problem of the diffusion equation with fractional view analysis. The suggested model is an important mathematical model to study the behavior of degrees of memory in diffusing materials. Some important results for the given model at different fractional orders of the derivatives are achieved. Graphs show the obtained results to confirm the accuracy and validity of the suggested technique. These results are in good contact with the physical dynamics of the targeted problems. The obtained results for both fractional and integer orders problems are explained through graphs and tables. Tables and graphs support the physical behavior of each problem and the best of physical analysis. From the results, it is concluded that as the fractional order derivative is changed, the graphs or paths of dynamics are also changed. Therefore, we now choose the best solution or dynamic of the problem at a particular derivative order. It is analyzed that the present technique is one of the best techniques to handle the solutions of fractional partial differential equations having initial and boundary conditions (BCs), which are very rare in literature. Furthermore, a small number of calculations are done to achieve a very high rate of convergence, which is the novelty of the present research work. The proposed method provides the series solution with twice recursive formulae to increase the desired accuracy and is preferred among the best techniques to find the solution of fractional partial differential equations with mixed initials and BCs.

Generalization of the Modified Bernstein Polynomials Method for Solving Fractional Coupled Equal Width Wave Equations

The objective of this research paper is to find the approximate solution to the fractional partial differential equations, which are difficult to find the exact solution, and therefore, by relying on the modified Bernstein polynomial, the study suggested generalizing the method to be available for both space and time fractional partial differential equations. In order to investigate the ability of the method to reach the approximate solution, the method was applied to the coupled of space-time-fractional of the equal width wave equations(FCEWE) using Caputo’s definition of the fractional derivative, the results of the application showed the ability of the method to find the approximate solution.

ARA-residual power series method for solving partial fractional differential equations

In this article a new approach in solving time fractional partial differential equations (TFPDEs) is introduced, that is, the ARA-residual power series method. The main idea of this technique, depends on applying the ARA-transform and using Taylor's expansion to construct approximate series solutions. The procedure of getting the approximate solutions for nonlinear TFPDEs is a difficult mission, the ARA-residual power series method over comes this trouble throughout expressing the solution in a series form then obtain the series coefficients using the idea of the residual function and the concept of the limit at infinity. This method is efficient and applicable to solve a wide family of TFPDEs. Four attractive applications are considered to show the speed and the strength of the proposed method in constructing solitary series solutions of the target equations.

ANALYTICAL SOLUTIONS OF (1+1)- DIMENSIONAL DISTRIBUTED LONG WAVE (DLW) EQUATION WITH AUXILIARY EQUATION METHOD

In this article, the exact solutions of the (1+1)-dimensional (DLW) equation, a fractional partial differential equation in conformable sense, which is a nonlinear, are given. Furthermore, with the aid of the mathematica program it is seen that the analytical solutions revealed with the auxiliary equation method satisfies the equation.