Fractional Partial Integro-differential Equations Research Articles

Uniqueness and Hyers–Ulam’s stability for a fractional nonlinear partial integro-differential equation with variable coefficients and a mixed boundary condition

Abstract Introducing a pair-parameter matrix Mittag–Leffler function, we study the uniqueness and Hyers–Ulam stability to a new fractional nonlinear partial integro-differential equation with variable coefficients and a mixed boundary condition using Banach’s contractive principle as well as Babenko’s approach in a Banach space. These investigations have serious applications since uniqueness and stability analysis are essential topics in various research fields. The techniques used also work for different types of differential equations with initial or boundary conditions, as well as integral equations. Moreover, we present a Python code to compute approximate values of our newly established pair-parameter matrix Mittag–Leffler functions, which extend the multivariate Mittag–Leffler function. A few examples are given to show applications of the key results obtained.

An algorithmic exploration of variable order fractional partial integro-differential equations via hexic shifted Chebyshev polynomials

An algorithmic exploration of variable order fractional partial integro-differential equations via hexic shifted Chebyshev polynomials

Analysis of Some Semi-analytical Methods for the Solutions of a Class of Time Fractional Partial Integro-differential Equations

Analysis of Some Semi-analytical Methods for the Solutions of a Class of Time Fractional Partial Integro-differential Equations

LADM procedure to find the analytical solutions of the nonlinear fractional dynamics of partial integro-differential equations

Abstract Generally, fractional partial integro-differential equations (FPIDEs) play a vital role in modeling various complex phenomena. Because of the several applications of FPIDEs in applied sciences, mathematicians have taken a keen interest in developing and utilizing the various techniques for its solutions. In this context, the exact and analytical solutions are not very easy to investigate the solution of FPIDEs. In this article, a novel analytical approach that is known as the Laplace adomian decomposition method is implemented to calculate the solutions of FPIDEs. We obtain the approximate solution of the nonlinear FPIDEs. The results are discussed using graphs and tables. The graphs and tables have shown the greater accuracy of the suggested method compared to the extended cubic-B splice method. The accuracy of the suggested method is higher at all fractional orders of the derivatives. A sufficient degree of accuracy is achieved with fewer calculations with a simple procedure. The presented method requires no parametrization or discretization and, therefore, can be extended for the solutions of other nonlinear FPIDEs and their systems.

Adaptive loss weighting auxiliary output fPINNs for solving fractional partial integro-differential equations

We propose an adaptive loss weighting auxiliary output fractional physics-informed neural networks (AWAO-fPINNs) based on the fractional physics-informed neural networks (fPINNs) for solving fractional partial integro-differential equations. In this framework, the automatic differentiation technique and numerical differentiation algorithm are effectively combined to construct a universal numerical scheme for Caputo fractional derivatives of different orders. Secondly, using the multi-output form of neural networks, the main output represents the required numerical solution and the auxiliary outputs represent the integrals in the equation. The relationships and new constraints between all outputs are obtained through automatic differentiation, successfully avoiding the discretization of the integrals. In addition, an adaptive loss weighting strategy is introduced into the model, which is based on the maximum likelihood estimation to continuously update the adaptive weights to realize the automatic allocation of loss weights, thus improving the accuracy of the model. Finally, we verify the effectiveness and accuracy of the AWAO-fPINNs model via several numerical experiments.

Application of a matrix Mittag-Leffler function to the fractional partial integro-differential equation in R n

Application of a matrix Mittag-Leffler function to the fractional partial integro-differential equation in R n

Remarks on a fractional nonlinear partial integro-differential equation via the new generalized multivariate Mittag-Leffler function

Introducing a new generalized multivariate Mittag-Leffler function which is a generalization of the multivariate Mittag-Leffler function, we derive a sufficient condition for the uniqueness of solutions to a brand new boundary value problem of the fractional nonlinear partial integro-differential equation using Banach’s fixed point theorem and Babenko’s technique. This has many potential applications since uniqueness is an important topic in many scientific areas, and the method used clearly opens directions for studying other types of equations and corresponding initial or boundary value problems. In addition, we use Python which is a high-level programming language efficiently dealing with the summation of multi-indices to compute approximate values of the generalized Mittag-Leffler function (it seems impossible to do so by any existing integral representation of the Mittag-Leffler function), and provide an example showing applications of key results derived.

Sinc Collocation Method to Simulate the Fractional Partial Integro-Differential Equation with a Weakly Singular Kernel

In this article, we develop an efficient numerical scheme for dealing with fractional partial integro-differential equations (FPIEs) with a weakly singular kernel. The weight and shift Grünwald difference (WSGD) operator is adopted to approximate a time fractional derivative and the Sinc collocation method is applied for discretizing the spatial derivative.The exponential convergence of our proposed method is demonstrated in detail. Finally, numerical evidence is employed to verify the theoretical results and confirm the expected convergence rate.

A Matrix Mittag–Leffler Function and the Fractional Nonlinear Partial Integro-Differential Equation in ℝn

In this paper, we introduce the matrix Mittag–Leffler function, which is a generalization of the multivariate Mittag–Leffler function, in order to investigate the uniqueness of the solutions to a fractional nonlinear partial integro-differential equation in Rn with a boundary condition based on Banach’s contractive principle and Babenko’s approach. In addition, we present an example demonstrating applications of the key results derived using a Python code that computes the approximate value of our matrix Mittag–Leffler function.

An efficient spline technique for solving time-fractional integro-differential equations

Spline curves are very prominent in the mathematics due to their simple construction, accuracy of assessment and ability to approximate complicated structures into interactive curved designs. A spline is a smooth piece-wise polynomial function. The primary goal of this study is to use extended cubic B-spline (ExCuBS) functions with a new second order derivative approximation to obtain the numerical solution of the weakly singular kernel (SK) non-linear fractional partial integro-differential equation (FPIDE). The spatial and temporal fractional derivatives are discretized by ExCuBS and the Caputo finite difference scheme, respectively. The present study found that it is stable and convergent. The validity of the current approach is examined on a few test problems, and the obtained outcomes are compared with those that have previously been reported in the literature.

An adaptive finite element method for Riesz fractional partial integro-differential equations

AbstractThe Riesz fractional derivative has been employed to describe the spatial derivative in a variety of mathematical models. In this work, the accuracy of the finite element method (FEM) approximations to Riesz fractional derivative was enhanced by using adaptive refinement. This was accomplished by deducing the Riesz derivatives of the FEM bases to work on non-uniform meshes. We utilized these derivatives to recover the gradient in a space fractional partial integro-differential equation in the Riesz sense. The recovered gradient was used as an a posteriori error estimator to control the adaptive refinement scheme. The stability and the error estimate for the proposed scheme are introduced. The results of some numerical examples that we carried out illustrate the improvement in the performance of the adaptive technique.

On the Uniqueness of the Bounded Solution for the Fractional Nonlinear Partial Integro-Differential Equation with Approximations

This paper studies the uniqueness of the bounded solution to a new Cauchy problem of the fractional nonlinear partial integro-differential equation based on the multivariate Mittag–Leffler function as well as Banach’s contractive principle in a complete function space. Applying Babenko’s approach, we convert the fractional nonlinear equation with variable coefficients to an implicit integral equation, which is a powerful method of studying the uniqueness of solutions. Furthermore, we construct algorithms for finding analytic and approximate solutions using Adomian’s decomposition method and recurrence relation with the order convergence analysis. Finally, an illustrative example is presented to demonstrate constructions for the numerical solution using MATHEMATICA.

Analytical and Numerical Solution for the Time Fractional Black-Scholes Model Under Jump-Diffusion

AbstractIn this work, we study the numerical solution for time fractional Black-Scholes model under jump-diffusion involving a Caputo differential operator. For simplicity of the analysis, the model problem is converted into a time fractional partial integro-differential equation with a Fredholm integral operator. The L1 discretization is introduced on a graded mesh to approximate the temporal derivative. A second order central difference scheme is used to replace the spatial derivatives and the composite trapezoidal approximation is employed to discretize the integral part. The stability results for the proposed numerical scheme are derived with a sharp error estimation. A rigorous analysis proves that the optimal rate of convergence is obtained for a suitable choice of the grading parameter. Further, we introduce the Adomian decomposition method to find out an analytical approximate solution of the given model and the results are compared with the numerical solutions. The main advantage of the fully discretized numerical method is that it not only resolves the initial singularity occurred due to the presence of the fractional operator, but it also gives a higher rate of convergence compared to the uniform mesh. On the other hand, the Adomian decomposition method gives the analytical solution as well as a numerical approximation of the solution which does not involve any mesh discretization. Furthermore, the method does not require a large amount of computer memory and is free of rounding errors. Some experiments are performed for both methods and it is shown that the results agree well with the theoretical findings. In addition, the proposed schemes are investigated on numerous European option pricing jump-diffusion models such as Merton’s jump-diffusion and Kou’s jump-diffusion for both European call and put options.

Solving Some Physics Problems Involving Fractional-Order Differential Equations with the Morgan-Voyce Polynomials

In this research, we present a new computational technique for solving some physics problems involving fractional-order differential equations including the famous Bagley–Torvik method. The model is considered one of the important models to simulate the coupled oscillator and various other applications in science and engineering. We adapt a collocation technique involving a new operational matrix that utilizes the Liouville–Caputo operator of differentiation and Morgan–Voyce polynomials, in combination with the Tau spectral method. We first present the differentiation matrix of fractional order that is used to convert the problem and its conditions into an algebraic system of equations with unknown coefficients, which are then used to find the solutions to the proposed models. An error analysis for the method is proved to verify the convergence of the acquired solutions. To test the effectiveness of the proposed technique, several examples are simulated using the presented technique and these results are compared with other techniques from the literature. In addition, the computational time is computed and tabulated to ensure the efficacy and robustness of the method. The outcomes of the numerical examples support the theoretical results and show the accuracy and applicability of the presented approach. The method is shown to give better results than the other methods using a lower number of bases and with less spent time, and helped in highlighting some of the important features of the model. The technique proves to be a valuable approach that can be extended in the future for other fractional models having real applications such as the fractional partial differential equations and fractional integro-differential equations.

Two unified families of bivariate Mittag-Leffler functions

The various bivariate Mittag-Leffler functions existing in the literature are gathered here into two broad families. Several different functions have been proposed in recent years as bivariate versions of the classical Mittag-Leffler function; we seek to unify this field of research by putting a clear structure on it. We use our general bivariate Mittag-Leffler functions to define fractional integral operators (which have a semigroup property) and corresponding fractional derivative operators (which act as left inverses and analytic continuations). We also demonstrate how these functions and operators arise naturally from some fractional partial integro-differential equations of Riemann–Liouville type.

An efficient numerical approach for solving variable-order fractional partial integro-differential equations

An efficient numerical approach for solving variable-order fractional partial integro-differential equations

An Efficient Approach to Approximate the Solutions of Fractional Partial Integro-Differential Equations

In this article a computational efficient approach is presented so as to examine the approximate solutions (AP) of fractional One-dimensional partial integro-differential equations (CF1DPDEs). The fractional derivative will be in the conformable sense. The suggested approach combined between the shifted Legendre polynomials and a semi-analytic approach. the proposed approach is tested by some examples which are introduced to illustrate its accuracy, applicability and efficiency.

Numerical approximation of time fractional partial integro-differential equation based on compact finite difference scheme

In this paper, a new numerical scheme based on weighted and shifted Grünwald formula and compact difference operate is proposed. The proposed numerical scheme is used to solve time fractional partial integro-differential equation with a weakly singular kernel. Meanwhile the time fractional derivative is denoted by the Riemann-Liouville sense. Subsequently, we prove the stability and convergence of the mentioned numerical scheme and show that the numerical solution converges to the analytical solution with order O(τ2 + h4), where τ and h are time step size and space step size, respectively. The advantage is that the accuracy of the suggested schemes is not dependent on the fractional α. Furthermore, the numerical example shows that the method proposed in this paper is effective, and the calculation results are consistent with the theoretical analysis.

Pell Collocation Method for Solving the Nonlinear Time–Fractional Partial Integro–Differential Equation with a Weakly Singular Kernel

This article focuses on finding the numerical solution of the nonlinear time–fractional partial integro–differential equation. For this purpose, we use the operational matrices based on Pell polynomials to approximate fractional Caputo derivative, nonlinear, and integro–differential terms; and by collocation points, we transform the problem to a system of nonlinear equations. This nonlinear system can be solved by the fsolve command in Matlab. The method’s stability and convergence have been studied. Also included are five numerical examples to demonstrate the veracity of the suggested strategy.

Numerical Analysis of Iterative Fractional Partial Integro-Differential Equations

Many nonlinear phenomena are modeled in terms of differential and integral equations. However, modeling nonlinear phenomena with fractional derivatives provides a better understanding of processes having memory effects. In this paper, we introduce an effective model of iterative fractional partial integro-differential equations (FPIDEs) with memory terms subject to initial conditions in a Banach space. The convergence, existence, uniqueness, and error analysis are introduced as new theorems. Moreover, an extension of the successive approximations method (SAM) is established to solve FPIDEs in sense of Caputo fractional derivative. Furthermore, new results of stability analysis of solution are also shown.