Fractional Partial Differential Equations Research Articles

Comparative analysis of fractional dynamical systems with various operators

<abstract><p>This article implements an efficient analytical technique within three different operators to investigate the solutions of some fractional partial differential equations and their systems. The generalized schemes of the proposed method are derived for every targeted problem under the influence of each fractional derivative operator. The numerical examples of the non-homogeneous fractional Cauchy equation and three-dimensional Navier-Stokes equations are obtained using the new iterative transform method. The obtained results under different fractional derivative operators are found to be identical. The 2D and 3D plots have confirmed the close connection between the exact and obtained results. Moreover, the table shows the higher accuracy of the proposed method.</p></abstract>

Conformable finite element method for conformable fractional partial differential equations

<abstract><p>The finite element (FE) method is a widely used numerical technique for approximating solutions to various problems in different fields such as thermal diffusion, mechanics of continuous media, electromagnetism and multi-physics problems. Recently, there has been growing interest among researchers in the application of fractional derivatives. In this paper, we present a generalization of the FE method known as the conformable finite element method, which is specifically designed to solve conformable fractional partial differential equations (CF-PDE). We introduce the basis functions that are used to approximate the solution of CF-PDE and provide error estimation techniques. Furthermore, we provide an illustrative example to demonstrate the effectiveness of the proposed method. This work serves as a starting point for tackling more complex problems involving fractional derivatives.</p></abstract>

Simulating two-dimensional optimal control problem of fractional partial differential equations

Simulating two-dimensional optimal control problem of fractional partial differential equations

Numerical analysis of fractional heat transfer and porous media equations within Caputo-Fabrizio operator

<abstract><p>This paper presents a comparative study of two popular analytical methods, namely the Homotopy Perturbation Transform Method (HPTM) and the Adomian Decomposition Transform Method (ADTM), to solve two important fractional partial differential equations, namely the fractional heat transfer and porous media equations. The HPTM uses a perturbation approach to construct an approximate solution, while the ADTM decomposes the solution into a series of functions using the Adomian polynomials. The results obtained by the HPTM and ADTM are compared with the exact solutions, and the performance of both methods is evaluated in terms of accuracy and convergence rate. The numerical results show that both methods are efficient in solving the fractional heat transfer and porous media equations, and the HPTM exhibits slightly better accuracy and convergence rate than the ADTM. Overall, the study provides a valuable insight into the application of the HPTM and ADTM in solving fractional differential equations and highlights their potential for solving complex mathematical models in physics and engineering.</p></abstract>

CONSTRUCTION OF CLOSED FORM SOLITON SOLUTIONS TO THE SPACE-TIME FRACTIONAL SYMMETRIC REGULARIZED LONG WAVE EQUATION USING TWO RELIABLE METHODS

The employment of nonlinear fractional partial differential equations (NLFPDEs) is not limited to branches of mathematics entirely but also applicable in other science fields such as biology, physics and engineering. This paper derives some solitary wave solutions for the space-time fractional symmetric regularized long wave (SRLW) equation by means of the improved [Formula: see text]-expansion approach and the [Formula: see text]-expansion method. We use the definition of the Jumarie’s modified Riemann–Liouville derivative to handle the fractional derivatives appearing in this equation. Diverse types of soliton solutions are successfully expressed on the form of rational, hyperbolic, trigonometric, and complex functions. We extract kink wave, internal solitary wave, and solitary wave solutions. The performances of the proposed methods are compared with each other. Moreover, we compare the constructed results with some published solutions. The long behaviors of the obtained solutions are plotted in 2D and 3D figures. The resulting outcomes point out that the used techniques promise to empower us to deal with more NLFPDEs arising in mathematical physics.

A hybrid analytical technique for solving multi-dimensional time-fractional Navier-Stokes system

<abstract> <p>In this research, a hybrid method, entitled the Laplace Residual Power Series technique, is adapted to find series solutions to a time-fractional model of Navier-Stokes equations in the sense of Caputo derivative. We employ the proposed method to construct analytical solutions to the target problem using the idea of the Laplace transform and the residual function with the concept of limit at infinity. A simple modification of the suggested method is presented to deal easily with the nonlinear terms constructed on the properties of the power series. Three interesting examples are solved and compared with the exact solutions to test the reliability, simplicity, and capacity of the presented method of solving systems of fractional partial differential equations. The results indicate that the used technique is a simple approach for solving nonlinear fractional differential equations since it depends only on the residual functions and the concept of the limit at infinity without needing differentiation or other complex computations.</p> </abstract>

NON-DIFFERENTIABLE EXACT SOLUTIONS OF THE LOCAL FRACTIONAL ZAKHAROV–KUZNETSOV EQUATION ON THE CANTOR SETS

In this study, a new fractional Zakharov–Kuznetsov equation (ZKE) within the local fractional derivative (LFD) is derived. Yang’s non-differentiable (ND) traveling wave transform is introduced, then two novel techniques namely the Mittag-Leffler function-based method (MLFBM) and Yang’s special function method (Y-SFM) are adopted to seek for the ND exact solutions for the first time. With the aid of the Mathematica software, the dynamic behaviors of the different solutions on the Cantor sets are illustrated via the 3D plots by assigning the appropriate parameters. The attained results confirm that the mentioned methods are effective and straightforward, which can be used to study the ND exact solutions of the local fractional partial differential equations (PDEs).

Comparison of two modified analytical approaches for the systems of time fractional partial differential equations

<abstract><p>The aim of this article is to present a comparison of two analytical approaches toward obtaining the solution of the time-fractional system of partial differential equations. The newly proposed approaches are the new approximate analytical approach (NAAA) and Mohand variational iteration transform approach (MVITA). The NAAA is based on the Caputo-Riemann operator and its basic properties with the decomposition procedure. The NAAA provides step wise series form solutions with fractional order, which quickly converge to the exact solution for integer order. The MVITA is based on a variational iteration procedure and uses the Mohand integral transform. The MVITA also provides a series solution without a stepwise solution. Both approaches provide a series form of solutions to the proposed problems. The analytical procedures and obtained results are compared for the proposed problems. The obtained results were also compared with exact solutions for the problems. The obtained result and plots have shown the validity and applicability of the proposed algorithms. Both approaches can be extended for the analytical solution of other physical phenomena in science and technology.</p></abstract>

Symmetry analysis and conservation laws of time fractional Airy type and other KdV type equations

<abstract><p>We study the invariance properties of the fractional time version of the nonlinear class of equations $ u_{t}^{\alpha}-g(u)\; u_{x}-f(u)\; u_{xxx} = 0 $, where $ 0 < \alpha < 1 $ using some recently developed symmetry-based techniques. The equations reduce to ordinary fractional Airy type, Korteweg-de Vries (KdV) and modified KdV equations through the change of variables provided by the symmetries. Furthermore, we utilize the symmetries to construct conservation laws for the fractional partial differential equations.</p></abstract>

Local meshless collocation scheme for numerical simulation of space fractional PDE

In this work, numerical solution of multi term space fractional PDE is calculated by using radial basis functions. The fractional derivatives of radial basis functions are evaluated by Caputo and Riemann-Liouville definitions. Local radial basis functions are applied to get stable and accurate solution the problem. Accuracy of the method is assessed by using double mesh procedure. Numerical solutions are presented for different fractional orders to show the effect of introducing fractionality.

SUMUDU TRANSFORM FOR TIME FRACTIONAL PHYSICAL MODELS AN ANALYTICAL ASPECT

Present paper deals with the development of a novel and reliable algorithm to solve various time-fractional differential prototypes regarding engineering and physics. The developed algorithm is named as Sumudu Iterative Transform Regime. In present work, proposed regime is applied to tackle different models of importance. The fetched results have shown the efficiency, efficacy and reliability of the developed scheme. In most of the cases, closed form of the solutions is provided. Moreover, profiles of solutions are provided to show the behavior of the fetched results. Error analysis of the results is already notified as well as convergence aspect is also mentioned. On the basis of the discussed aspects, it can be claimed the Sumudu Iterative Transform Regime is a robust technique to deal with the complex natured PDEs. Present scheme will surely add importance in the literature. With the aid of the present regime numerous fractional PDEs and partial-integro differential equations can be tackled.

NEW EXACT SOLUTIONS OF THE LOCAL FRACTIONAL MODIFIED EQUAL WIDTH-BURGERS EQUATION ON THE CANTOR SETS

This study proposes a new fractal modified equal width-Burgers equation (MEWBE) with the local fractional derivative (LFD) for the first time. By defining the Mittag-Leffler function (MLF) on the Cantor set (CS), two special functions, namely, the [Formula: see text] and [Formula: see text] functions, are derived for constructing the auxiliary function to seek the non-differentiable (ND) exact solutions. And 16 groups of the ND exact solutions are successfully established. The solutions on the CS are depicted graphically to interpret the nonlinear dynamic behaviors. Furthermore, the comparative results of the fractal MEWBE and the classical MEWBE are also discussed. The obtained results confirm that the proposed method is effective and powerful, and can provide a promising way to find the ND exact solutions of the local fractional PDEs.

N-SOLITON, BREATHER, LUMP SOLUTIONS AND DIVERSE TRAVELING WAVE SOLUTIONS OF THE FRACTIONAL (2 + 1)-DIMENSIONAL BOUSSINESQ EQUATION

The [Formula: see text]-dimensional Boussinesq equation plays a key role in modeling the shallow water. In this work, we derive a new fractional [Formula: see text]-dimensional Boussinesq equation based on the conformable fractional derivative for the first time. By means of the Hirota bilinear method, we obtain the [Formula: see text]-soliton, breather and lump solutions. In addition, the abundant traveling wave solutions like bright solitary, dark solitary wave solutions are investigated by applying the variational method. The solutions are presented through the 3D plots and 2D contours by assigning the proper parameters. The corresponding physical interpretations are also elaborated. The findings in this work are expected to open some new horizons on the study of fractional PDEs in physics.

Solving Some Partial Differential Equations by Using Double Laplace Transform in the Sense of Nonconformable Fractional Calculus

In this paper, we introduce the non-conformable double Laplace transform. Its properties are studied, and it is applied to solve some fractional PDEs involving the nonconformable fractional derivative. Graphical representations of the obtained solutions are shown in figures. The study shows that this transform is effective and easy to apply to create an exact solution for types of fractional PDEs.

Computational analysis of local fractional partial differential equations in realm of fractal calculus

In this paper, a hybrid local fractional technique is applied to some local fractional partial differential equations. Partial differential equations modeled with local fractional derivatives easily capture the behavior of fractal models. The present technique is a copulation of local fractional homotopy method and local fractional integral transform. Four examples are provided to show the efficiency of an implemented method. Furthermore, computer simulations have also been performed for all the four examples of local fractional partial differential equations in a fractal domain. The working procedure depicts that the applied technique is very useful to acquire solutions for given local fractional partial differential equations in an efficient way. Moreover, the obtained solutions are also in good agreement with solutions computed by other methods.

Forced oscillation of impulsive fractional partial differential equations

In this paper, we investigate the forced oscillation of impulsive fractional partial differential equations. Some sufficient conditions are established which guarantee that the oscillation of the solutions by applying the integral transformation technique and differential inequality method. Also an example is shown to illustrate the effectiveness of our main results.

A computational numerical algorithm for thermal characterization of fractional unsteady free convection flow in an open cavity

A significant problem to study is how the fractional operators and physical structures may be interrelated and interconnected. The current model study reports the fractional time-dependent viscous, electric conductive fluid between two permeable infinitely large walls. The flow is driven by the mutual actions of imposed thermal buoyancy, pressure gradient, and transverse magnetic field of uniform strength. The suction and injection of the fluid take place at the right and left walls respectively. Transformations are used to convert the physical model into equivalent fractional partial differential equations (FPDEs). The finite difference computational code based on three difference fractional operators is developed to seek the behavior of modeled problem. The analysis of the model and code endorsement is provided through set of graphical plots and tabular form. It is noted that an increment into the Hartmann and Reynolds number causes a dropped pattern of the velocity while the drop is more substantial for the smaller choices of fractional parameters. Higher choices of heat source, thermal radiation, Ecker, pressure gradient and magnetic parameters cause an enhanced behavior of the thermal profile. The smaller values caused a slight increment on the thermal layer as higher choices of the fractional parameters. However, the patterns of the thermal and velocity layers are found clearer while using the ABC and CF fractional operators compared with the CC idea of fractional derivative.

Effective Optimized Decomposition Algorithms for Solving Nonlinear Fractional Differential Equations

Abstract In this paper, the optimized decomposition method, which was developed to solve integer-order differential equations, will be modified and extended to handle nonlinear fractional differential equations. Fractional derivatives will be considered in terms of Caputo sense. The suggested modifications design new optimized decompositions for the series solutions depending on linear approximations of the nonlinear equations. Two optimized decomposition algorithms have been introduced to obtain approximate solutions of broad classes of initial value problems (IVPs) consisting of nonlinear fractional ordinary differential equations (ODEs) and partial differential equations (PDEs). A comparative study was conducted between the proposed algorithms and the Adomian decomposition method (ADM) by means of some test illustration problems. The implemented numerical simulation results showed that the proposed algorithms give better accuracy and convergence, and reduce the complexity of computational work compared to the Adomian's approach. This confirms the belief that the optimized decomposition method will be used effectively and widely as a powerful tool in solving various fractional differential equations.

Analytical Approaches for Approximate Solution of the Time-Fractional Coupled Schrödinger–KdV Equation

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.

New soliton solutions and modulation instability analysis of fractional Huxley equation

In this research, the new auxiliary equation method (NAEM) for higher order nonlinear fractional Huxley equation is being employed to extricate the novel soliton solutions using Beta and M-Truncated fractional derivatives. For waves of finite amplitude, the Huxley equation demonstrates a substantial transfer of spectrum energy. A comparison of the solutions of the model with both fractional derivatives is also included in this research. Various kinds of solitary traveling wave solutions, such as trigonometric, hyperbolic, exponential, rational functions, etc., are found. These types of solutions demonstrate the superiority of the novelty of the method. This method’s key advantage over others is that it provides more broad solutions with certain flexible parameters. 3D and 2D graphs are used graphically to demonstrate the dynamical structures of the solutions. The results are presented in a way that demonstrate the usefulness and competence of the approach used to handle various nonlinear fractional partial differential equations. Lastly, we investigate the comparison of the gain spectra for modulation instability and the depiction of certain noteworthy outcomes by illustratively depicting the 2D figures produced by carefully considering the parameters