Partial Differential Equations Of Fractional Order Research Articles

Impact of fractional and integer order derivatives on the [formula omitted]-dimensional fractional Davey–Stewartson–Kadomtsev–Petviashvili equation

In this study, the closed-form wave solutions of the (4+1)-dimensional fractional Davey–Stewartson–Kadomtsev–Petviashvili equation are investigated using the modified auxiliary equation method and the Jacobi elliptic function method. In the analysis, two fractional derivatives known as M-truncated, beta and integer order derivative are used. The fractional-order partial differential equation is transformed into an integer-order ordinary differential equation by using the wave transformation, fractional derivatives, and integer-order derivatives. As a result, wave function solutions are found, including bell shape, W-shaped, composite dark-bright and periodic wave. The effects of free parameters on the amplitudes and wave behaviors are illustrated. It is demonstrated extensively that changes in the free parameters lead to changes in the wave amplitude. A comparison of solutions using the two types of fractional derivatives and the integer-order derivatives is included. The effects of the beta derivative, the M-truncated derivative and integer order derivative on the considered model are presented using 2D and 3D figures.

Combining approach of collocation and finite difference methods for fractional parabolic PDEs

This research aims to estimate the solutions of fractional-order partial differential equations of spacial fractional and both time-space fractional order. For this, we use finite differences for time derivatives and the well-known collocation method for space derivatives with lower-order Bernstein polynomials as basis functions. We explain the mathematical formulations in detail. Convergence and stability analysis of the space–time fractional diffusion equation with the source term is reported subsequently. Three numerical examples are considered for demonstrating the accuracy and reliability of the proposed method.

Analytical solutions fractional order partial differential equations arising in fluid dynamics

Analytical solutions fractional order partial differential equations arising in fluid dynamics

Application of the Sumudu Variational Iteration Method with Atangana-Baleanu-Caputo Operator for Solving Fractional-Order Heat-Like Equations with Initial Conditions

Fractional calculus techniques are widely utilized across various engineering disciplines and applied sciences. Among these techniques is the Sumudu Variational Iteration Method (SVIM), which has not yet been tested with the Atangana-Baleanu-Caputo fractional derivative in academic literature. This work aims to explore the application of SVIM for solving fractional-order partial differential equations using the Atangana-Baleanu-Caputo derivative. The method integrates the Sumudu transform with the variational iteration method. To demonstrate the effectiveness and validity of SVIM, we apply it to solve one-dimensional (1-D), two-dimensional (2-D), and three-dimensional (3-D) fractional-order heat-like partial differential equations. The results indicate that SVIM is both convergent and efficient for solving these types of fractional partial differential equations.

The Double Inversion Technique in the Some Blaise Abbo Method Applied to Schrodinger-Type Fractional-Order Evolution Equations

In this article we have used the new double inversion technique in the Some Blaise Abbo(SBA) method to solve some fractional order partial differential equations (edp) in the Caputo sense of the Schrodinger type by integrating all boundary conditions.

Study of [formula omitted] dimensional fractional order non-linear Benney equation using an analytical technique

The area related to fractional order partial differential equations (FOPDEs) has also attracted attention. Therefore, we focuss our attention to study nonlinear Benney equations of using power law type Caputo derivative of fractional order. To compute a series-type solution for the problem under consideration, we combine the Adomian approach with the Laplace transform. Our findings are supported by graphical representations and numerical interpretations, illustrating the applicability and insights gained from this approach in solving complex real-world problems.

An efficient collocation technique based on operational matrix of fractional-order Lagrange polynomials for solving the space-time fractional-order partial differential equations

In this research, we propose a novel and fast computational technique for solving a class of space-time fractional-order linear and non-linear partial differential equations. Caputo-type fractional derivatives are considered. The proposed method is based on the operational and pseudo-operational matrices for the fractional-order Lagrange polynomials. To carry out the method, first, we find the integer and fractional-order operational and pseudo-operational matrix of integration. The collocation technique and obtained operational and pseudo-operational matrices are then used to generate a system of algebraic equations by reducing the given space-time fractional differential problem. The resultant algebraic system is then easily solved by Newton's iterative methods. The upper bound of the fractional-order operational matrix of integration is also provided, which confirms the convergence of fractional-order Lagrange polynomial's approximation. Finally, some numerical experiments are conducted to demonstrate the applicability and usefulness of the suggested numerical scheme.

Nonlinear dynamic wave characteristics of optical soliton solutions in ion-acoustic wave

Traveling wave solutions are utilized to depict reaction-diffusion and analyze electrical signal transmission and propagation in space-time nonlinear fractional order partial differential equations like the space-time fractional Telegraph and Kolmogorov-Petrovsky Piskunov equations, and that are used in physical science to model combustion, biological research to model nerve impulse propagation, chemical dynamics to model concentration in order wave propagation, and plasma to model the progression of a set of duffing oscillators. In this study, the new generalized (G′/G)−expansion technique was employed to construct some novel and more universal closed-form traveling wave solutions in the sense of conformable derivatives which explain the above-stated phenomena properly. By utilizing complex fractional transformation, the ordinary differential equations are generated from fractional order differential equations. The recommended technique allowed us to produce some dynamical wave patterns of kink, single soliton, compacton, periodic shape, multiple periodic waves, anti-kink, and other structures are developed, which are shown using 3D plots and contour plots to illustrate the physical layout clearly. The traveling waveform responses can be defined in terms of functions based on trigonometry, hyperbolic operations, and rational functions and that are quick, flexible, and simple to reproduce. Furthermore, the obtained closed-form solutions for nonlinear fractional evolution equations make stability analysis and accuracy comparison amongst numerical solvers easier, which asserts that the new generalized (G′/G)−expansion technique is one of the most proficient and effective approaches.

An Efficient Model for Solving Fractional Order Partial Differential Equations

To solve fractional order differential equations, a hybrid technique named “the natural Adomian decomposition method (NADM) with Caputo fractional operator (CFO)” has been applied. The combination approach incorporates both the natural transform method (NTM) and the Adomian decomposition method (ADM). Two challenges are overcome to validate and demonstrate the efficacy of the current process. It is also shown that the results acquired using the suggested technique are extremely like those obtained using other strategies. For a range of science and engineering difficulties, the proposed solution has been shown to be efficient, dependable, and simple to implement.

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>

An Easy-To-Use Tool to Solve Differential Equations with the Fractional Laplacian

Numerical simulation of fractional-order partial differential equations is a challenging task and the majority of computing environments does not provide support for these problems. In this paper we describe how to exploit some of the Matlab features (a programming language not supporting fractional calculus in a naive way) to solve partial differential equations with the spectral fractional Laplacian. For shortness we focus on fractional Poisson equations but the proposed approach can be extended, with just some technical difficulties, to more involved problems. This approach cannot be considered as a highly efficient and accurate way to solve fractional partial differential equations, but as an easy-to-use tool for non specialists in numerical computation to obtain solutions without having to produce sophisticated numerical codes.

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.

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.