Partial Differential Equations Of Fractional Order Research Articles

An Efficient Model for Solving Fractional Order Partial Differential Equations

An Efficient Model for Solving Fractional Order Partial Differential Equations

Comparative study of fractional Newell–Whitehead–Segel equation using optimal auxiliary function method and a novel iterative approach

This research explores the solution of the time-fractional Newell–Whitehead–Segel equation using two separate methods: the optimal auxiliary function method and a new iterative method. The Newell–Whitehead–Segel equation holds significance in modeling nonlinear systems, particularly in delineating stripe patterns within two-dimensional systems. Employing the Caputo fractional derivative operator, we address two case study problems pertaining to this equation through our proposed methods. Comparative analysis between the numerical results obtained from our techniques and an exact solution reveals a strong alignment. Graphs and tables illustrate this alignment, showcasing the effectiveness of our methods. Notably, as the fractional orders vary, the results achieved at different fractional orders are compared, highlighting their convergence toward the exact solution as the fractional order approaches an integer. Demonstrating both interest and simplicity, our proposed methods exhibit high accuracy in resolving diverse nonlinear fractional order partial differential equations.

Parameter Identification of Fractional Order Partial Differential Equation Model Based on Polynomial–Fourier Method

Parameter Identification of Fractional Order Partial Differential Equation Model Based on Polynomial–Fourier Method

An optimization method for solving a general class of the inverse system of nonlinear fractional order PDEs

In this paper, we introduce a general class of the inverse system of nonlinear fractional order partial differential equations (GCISNF-PDEs) with initial-boundary and two overdetermination conditions. An optimization method is considered based on the generalized shifted Legendre polynomials (GSLPs) for solving GCISNV-FPDEs. The concept of the fractional order derivatives (F-Ds) is utilized in the Caputo type. Operational matrices (OMs) of classical derivatives and F-Ds of GSLPs are extracted. Making use of GSLPs, OMs, and Lagrange multipliers method, we reduce the given GCISNF-PDEs into an algebraic system of equations. The proposed approach achieves satisfactory results simply for a small number of the novel GSLPs. In this work, two mathematical examples are illustrated to analyse the introduced method convergence and test its validity as well as applicability.

A Novel Numerical Method for Solving Nonlinear Fractional-Order Differential Equations and Its Applications

In this article, a considerably efficient predictor-corrector method (PCM) for solving Atangana–Baleanu Caputo (ABC) fractional differential equations (FDEs) is introduced. First, we propose a conventional PCM whose computational speed scales with quadratic time complexity O(N2) as the number of time steps N grows. A fast algorithm to reduce the computational complexity of the memory term is investigated utilizing a sum-of-exponentials (SOEs) approximation. The conventional PCM is equipped with a fast algorithm, and it only requires linear time complexity O(N). Truncation and global error analyses are provided, achieving a uniform accuracy order O(h2) regardless of the fractional order for both the conventional and fast PCMs. We demonstrate numerical examples for nonlinear initial value problems and linear and nonlinear reaction-diffusion fractional-order partial differential equations (FPDEs) to numerically verify the efficiency and error estimates. Finally, the fast PCM is applied to the fractional-order Rössler dynamical system, and the numerical results prove that the computational cost consumed to obtain the bifurcation diagram is significantly reduced using the proposed fast algorithm.

An improved approximate method for solving two-dimensional time-fractional-order Black-Scholes model: a finite difference approach

<abstract><p>In this paper, we considered the two-dimensional fractional-order Black-Scholes model in the Liouville-Caputo sense. The Black-Scholes model was an important tool in the financial market, used for determining option prices in the European-style market. However, finding a closed-form analytical solution for the fractional-order partial differential equation was challenging. To address this, we introduced an improved finite difference method for approximating the solution of the two-dimensional fractional-order Black-Scholes model in the Liouville-Caputo sense, based on the Crank-Nicolson finite difference method. This method combined the concepts of the finite difference method for solving the multidimensional Black-Scholes model and the finite difference method for solving the fractional-order heat equation. We analyzed the conditional stability and the order of convergence. Furthermore, numerical examples were provided to illustrate the determination of option prices.</p></abstract>

New applications of the fractional derivative to extract abundant soliton solutions of the fractional order PDEs in mathematics physics

The motive of this research work is to unravel the mysteries of nature through fractional-order partial differential equations (PDEs). Here, we focus on two important fractional order nonlinear PDEs, namely the fractional order (4+1)-dimensional Fokas equation, which is used to give the model of many physical phenomena and dynamical processes, and the other one is the fractional order (2+1)-dimensional breaking soliton equation which is used to analyze the nonlinear problems like optical fiber communications, ocean engineering, etc. Recently, it has been an essential topic to extract the new soliton solutions which are used to investigate the hidden physical conditions of the nonlinear fractional PDEs. So, it is essential to solve those nonlinear fractional PDEs which have a physical impact in the fields of science and modern engineering. In our investigation, we attempt to provide nonlinear wave propagation patterns and investigate the equations, as mentioned earlier, through a computational method. A computing operating software called Mathematica has been applied to get a clear visualization of our gained outcomes, and we ascertain such types of shapes as the bell shape soliton, the anti-bell shape soliton, the singular bell shape soliton, the periodic solution, and the singular periodic solution. Our obtained results can keep an indispensable role in explaining various physical phenomena of nature shortly, and the applied method is the very cogent, efficient, and interesting to extract such types of solutions. Since we extract abundant solutions of these models, so we hope that this article is the best applications of mention method.

Employing a Fractional Basis Set to Solve Nonlinear Multidimensional Fractional Differential Equations

Fractional-order partial differential equations have gained significant attention due to their wide range of applications in various fields. This paper employed a novel technique for solving nonlinear multidimensional fractional differential equations by means of a modified version of the Bernstein polynomials called the Bhatti-fractional polynomials basis set. The method involved approximating the desired solution and treated the resulting equation as a matrix equation. All fractional derivatives are considered in the Caputo sense. The resulting operational matrix was inverted, and the desired solution was obtained. The effectiveness of the method was demonstrated by solving two specific types of nonlinear multidimensional fractional differential equations. The results showed higher accuracy, with absolute errors ranging from 10−12 to 10−6 when compared with exact solutions. The proposed technique offered computational efficiency that could be implemented in various programming languages. The examples of two partial fractional differential equations were solved using Mathematica symbolic programming language, and the method showed potential for efficient resolution of fractional differential equations.

On the solutions of some nonlinear fractional partial differential equations using an innovative and direct procedure

In this article, a highly effective technique is implemented to obtain the approximate solutions of strongly nonlinear fractional order partial differential equations (NFPDEs). The findings of this study show the successful behavior of the fractional novel analytical method (FNAM), which can be used successfully for the solutions of common, severe NFPDEs. In the proposed method, the nonlinearity in each mathematical model is directly handled by using fractional Taylor series, which reduces the calculation effort. In this work, the method's strength is primarily demonstrated on NFPDEs, and the obtained results are displayed via graphs and tables. From the numerical simulations, it is evident that the suggested technique has greater accuracy despite smaller calculations. It is the most straightforward method for determining the formulaic solution to any type of NFPDE and is considered to be the unique numerical methodology.

Solutions of Time Fractional (1 + 3)-Dimensional Partial Differential Equations by the Natural Transform Decomposition Method (NTDM)

The current study employs the natural transform decomposition method (NTDM) to test fractional-order partial differential equations (FPDEs). The present technique is a mixture of the natural transform method and the Adomian decomposition method. For the purpose of checking the precis of our technique, some examples are offered, and the series solutions of these equations are introduced by using NTDM. The outcome shows that the suggested approach is very active and straightforward for obtaining a series solutions of FPDEs and is more accurate if we compare it with existing methods.

Retracted: Exact Solutions of Three Types of Conformable Fractional-Order Partial Differential Equations.

[This retracts the article DOI: 10.1155/2022/5295115.].

MHD Free Convection Flows for Maxwell Fluids over a Porous Plate via Novel Approach of Caputo Fractional Model

The ultimate goal of the article is the analysis of free convective flow of an MHD Maxwell fluid over a porous plate. The study focuses on understanding the dynamics of fluid flow over a moving plate in the presence of a magnetic field, where the magnetic lines of force can either be stationary or in motion along the plate. Further, we will investigate the heat and mass transfer characteristics of the system under specific conditions: constant species and thermal conductivity as functions of time. The study involves a symmetric temperature distribution that provides heat on both sides of the plane. Our analysis includes the study of the model for different instances of plate motion and variations in temperature. The fluid dynamics of the system are mathematically described using a system of fractional-order partial differential equations. To make the model independent of geometric units, dimensionless variables are introduced. Moreover, we employ the concept of fractional-order derivative operators in the sense of Caputo, which introduces a fractional dimension to the equations. Additionally, the integral Laplace transform and numerical algorithms are utilized to solve the problem. Finally, by using graphical analysis the contribution of physical parameters on the fluid dynamics is discussed and valuable findings are documented.

Analysis of chaotic structures, bifurcation and soliton solutions to fractional Boussinesq model

In this work, we used the space-time fractional coupled Boussinesq (STFCB) model that is essential tools in the study of quantum optics, steady physics, the variational string’s acoustic waves, ion vibrational frequencies, hydro-magnetic waves in cold plasma and many other fields. In order to put such new precise solutions of the aforementioned model, the modified Sardar-sub equation (MSSE) technique has been suggested with inside the sense of conformable derivative and the fractional order partial differential equation that is capable of changing into an ordinary differential equation by using the travelling wave transform. The scoring of solitons and other solutions demonstrates the MSSE technique compatibility for different constant values, which are shown in 3-D, 2-D and contour plots. Additionally, we discussed the examined model chaotic and dynamical tendencies. The theory of plane dynamical system is used to examine the chaotic patterns of the systems. The investigations are novel and unexamined. They can be utilized to explain the physical phenomena which have been simulated to provide details on the brief dynamical characteristics. According to numerical simulations modifying the parameters of frequencies and amplitudes has an impact on the system of dynamical properties. We indicated that the MSSE technique for creating precise solutions offers new and significant mathematical tools in applied mathematics.

A Fractional Analysis of Zakharov–Kuznetsov Equations with the Liouville–Caputo Operator

In this study, we used two unique approaches, namely the Yang transform decomposition method (YTDM) and the homotopy perturbation transform method (HPTM), to derive approximate analytical solutions for nonlinear time-fractional Zakharov–Kuznetsov equations (ZKEs). This framework demonstrated the behavior of weakly nonlinear ion-acoustic waves in plasma containing cold ions and hot isothermal electrons in the presence of a uniform magnetic flux. The density fraction and obliqueness of two compressive and rarefactive potentials are depicted. In the Liouville–Caputo sense, the fractional derivative is described. In these procedures, we first used the Yang transform to simplify the problems and then applied the decomposition and perturbation methods to obtain comprehensive results for the problems. The results of these methods also made clear the connections between the precise solutions to the issues under study. Illustrations of the reliability of the proposed techniques are provided. The results are clarified through graphs and tables. The reliability of the proposed procedures is demonstrated by illustrative examples. The proposed approaches are attractive, though these easy approaches may be time-consuming for solving diverse nonlinear fractional-order partial differential equations.

Homotopy Analysis Method using Jumarie’s Approach for Multi-Dimensional Nonlinear Schrödinger Equations

In this paper, we suggest a fractional functional for the Homotopy Analysis Method (HAM) to solve the nonlinear fractional order partial differential equations with fractional order initial conditions by using the modified Riemann–Liouville fractional derivative proposed by G. Jumarie. Graphs have been plotted for different values of α , which clearly reflect the reliability of proposed scheme.

The Fibonacci wavelets approach for the fractional Rosenau–Hyman equations

In this study, a novel approach, called the Fibonacci wavelet collocation technique (FWCT), is presented for the numerical solution of nonlinear fractional order partial differential equations (FPDEs). The chosen numerical strategy is based on the collocation method with the Fibonacci wavelet and its functional integration matrix. The newly developed numerical approach for the nonlinear time-fractional Rosenau–Hyman equation is the focus of this paper. The corresponding fractional order derivative is taken in the Caputo sense. This effective numerical approach converts the nonlinear problem into a set of algebraic-type nonlinear equations. This nonlinear system of equations is analyzed using the Newton–Raphson method to find the unknown coefficients. We provide the different types of error analysis of the present technique. The exact solution and the results obtained by the other literary methods are compared to the results achieved. The numerical outcomes demonstrate that the current analysis offers a more accurate approximation than the prior approaches. The result derived with the aid of this technique is in good agreement and indicates that the approach is easy to implement and accurate. These results reveal that the proposed method is computationally attractive, efficient, effective, reliable, and robust for solving various physical models in science and engineering. We made all calculations and plotted the graphs using Mathematica software.

Numerical solution of two‐dimensional nonlinear Riesz space‐fractional reaction–advection–diffusion equation using fast compact implicit integration factor method

AbstractIn the present article, a finite domain is considered to find the numerical solution of a two‐dimensional nonlinear fractional‐order partial differential equation (FPDE) with Riesz space fractional derivative (RSFD). Here two types of FPDE–RSFD are considered, the first one is a two‐dimensional nonlinear Riesz space‐fractional reaction–diffusion equation (RSFRDE) and the second one is a two‐dimensional nonlinear Riesz space‐fractional reaction‐advection‐diffusion equation (RSFRADE). SFRDE is obtained by simply replacing second‐order derivative term of the standard nonlinear diffusion equation by the Riesz fractional derivative of order whereas the SFRADE is obtained by replacing the first‐order and second‐order space derivatives from the standard order advection–dispersion equation with the Riesz fractional derivatives of order . A numerical method is provided to deal with the RSFD with the weighted and shifted Grünwald–Letnikov (WSGD) approximations, for the spatial discretization. The SFRDE and SFRADE are transformed into a system of ordinary differential equations (ODEs), which have been solved using a fast compact implicit integration factor (FcIIF) with nonuniform time meshes. Finally, the demonstration of the validation and effectiveness of the numerical method is given by considering some existing models.

Fractional Study of the Non-Linear Burgers’ Equations via a Semi-Analytical Technique

Most complex physical phenomena are described by non-linear Burgers’ equations, which help us understand them better. This article uses the transformation and the fractional Taylor’s formula to find approximate solutions for non-linear fractional-order partial differential equations. Solving non-linear Burgers’ equations with the right starting data shows that the method utilized is correct and can be utilized. Based on the limit of the idea, a rapid convergence McLaurin series is used to obtain close series solutions for both models with less work and more accuracy. To see how time-Caputo fractional derivatives affect how the results of the above models behave, in three dimension figures are drawn. The results showed that the proposed method is an easy, flexible, and helpful way to solve and understand a wide range of non-linear physical models.

Analysis and Numerical Simulation of System of Fractional Partial Differential Equations with Non-Singular Kernel Operators

The exact solution to fractional-order partial differential equations is usually quite difficult to achieve. Semi-analytical or numerical methods are thought to be suitable options for dealing with such complex problems. To elaborate on this concept, we used the decomposition method along with natural transformation to discover the solution to a system of fractional-order partial differential equations. Using certain examples, the efficacy of the proposed technique is demonstrated. The exact and approximate solutions were shown to be in close contact in the graphical representation of the obtained results. We also examine whether the proposed method can achieve a quick convergence with a minimal number of calculations. The present approaches are also used to calculate solutions in various fractional orders. It has been proven that fractional-order solutions converge to integer-order solutions to problems. The current technique can be modified for various fractional-order problems due to its simple and straightforward implementation.

Meshfree numerical approach for some time-space dependent order partial differential equations in porous media

<abstract><p>In this article, the meshfree radial basis function method based on the Gaussian function is proposed for some time-space dependent fractional order partial differential equation (PDE) models. These PDE models have significant applications in chemical engineering and physical science. Some main advantages of the proposed method are that it is easy to implement, and the output response is quick and highly accurate, especially in the higher dimension. In this method, the time-dependent derivative terms are treated by Caputo fractional derivative while space-dependent derivative terms are treated by Riesz, Riemann-Liouville, and Grünwald-Letnikov derivatives. The proposed method is tested on some numerical examples and the accuracy is analyzed by $ \|L\|_\infty $.</p></abstract>