Solving mixed classical and fractional partial differential equations using short–memory principle and approximate inverses

In this article, we focus on a class of singular linear systems of fractional differential equations with given nonconsistent initial conditions (IC). Because the nonconsistency of the IC can not lead to a unique solution for the singular system, we use two optimization techniques to provide an optimal solution for the system. We use two optimization techniques to provide the optimal solution for the system because a unique solution for the singular system cannot be obtained due to the non‐consistency of the IC. These two optimization techniques involve perturbations to the non‐consistent IC, specifically, an l2 perturbation (which seeks an optimal solution for the system in terms of least squares), and a second‐order optimization technique at an l1 minimum perturbation, (which includes an appropriate smoothing). Numerical examples are given to justify our theory. We use the Caputo (C) fractional derivative and two recently defined alternative versions of this derivative, the Caputo‐Fabrizio (CF) and the Atangana‐Baleanu (AB) fractional derivative.

This paper studies fractional differential equations (FDEs) with mixed fractional derivatives. Existence, uniqueness, stability, and asymptotic results are derived.

In this paper, a Haar wavelets based numerical method to solve a system of linear or nonlinear fractional differential equations has been proposed. Numerous nontrivial test examples along with practical problems from fluid dynamics and chemical engineering have been considered to illustrate applicability of the proposed method. We have derived a theoretical error bound which plays a crucial role whenever the exact solution of the system is not known and also it guarantees the convergence of approximate solution to exact solution.

Analytic study on linear systems of fractional differential equations

The fractional partial differential equation is an equation with non-integer partial derivative which is now widely used in solving various problems so that in recent years, many researchers are interested to study fractional partial differential equation. The equation which is used in this paper is the non-homogeneous fractional partial differential equation. The aim of this study is to determine general solution of the partial differential equation through three different order equations (α; 1), (β; 2) and (α; β), where 0 < α ≤ 2 and 0 < β ≤ 1 by using Homotopy Analysis Method (HAM). Furthermore, by taking a convergent sequence, it will be analyzed that the convergence of the sequence of differential equations results in the sequence of solutions of partial fractional non-homogeneous partial differential equations converging to the solution function of the fractional partial differential equation.

A novel Petrov-Galerkin method for a class of linear systems of fractional differential equations

System of fractional differential algebraic equations with applications

In the current study, an approximate scheme is established for solving the fractional partial differential equations (FPDEs) with Volterra integral terms via two‐dimensional block‐pulse functions (2D‐BPFs). According to the definitions and properties of 2D‐BPFs, the original problem is transformed into a system of linear algebra equations. By dispersing the unknown variables for these algebraic equations, the numerical solutions can be obtained. Besides, the proof of the convergence of this system is given. Finally, several numerical experiments are presented to test the feasibility and effectiveness of the proposed method.

We consider a system of Riemann-Liouville fractional partial differential equations with constant coefficients and obtain a general representation of solutions in a rectangular domain. The asymptotic behavior and other properties of the fundamental solution are studied.

Power Series Solution of Time-Fractional Majda-Biello System Using Lie Group Analysis

In this paper, we consider a time-fractional coupled KdV equations describing the interaction of equatorial and midlatitude Rossby waves. From the application of Lie group analysis, the governing system of fractional partial differential equations (FPDEs) is reduced to a system of fractional ordinary differential equations (FODEs). Further, we construct the group-invariant solution as well as the power series solution for the given coupled equations. Next, the evolutionary behavior of the waves under the influence of the fractional order derivative [Formula: see text] is studied graphically through the group-invariant solution. Finally, the conservation laws for the given system of FPDEs are obtained.

In this paper, the existence and uniqueness of solutions to a coupled formally symmetric system of fractional differential equations with nonlinear p-Laplacian operator and nonlinear fractional differential-integral boundary conditions are obtained by using the matrix eigenvalue method. The Hyers–Ulam stability of the coupled formally symmetric system is also presented with certain growth conditions. By using the topological degree theory and nonlinear functional analysis methods, some sufficient conditions for the existence and uniqueness of solutions to this coupled formally symmetric system are established. Examples are provided to verify our results.

In this paper, we discuss by means of a fixed point theorem, the existence of positive solutions of a system of nonlinear Caputo fractional differential equations with integral boundary conditions. An example is given to illustrate the main results.

We consider the existence and uniqueness of positive solution to nonzero boundary values problem for a coupled system of fractional differential equations. The differential operator is taken in the standard Riemann‐Liouville sense. By using Banach fixed point theorem and nonlinear differentiation of Leray‐Schauder type, the existence and uniqueness of positive solution are obtained. Two examples are given to demonstrate the feasibility of the obtained results.

In this article, MahgoubAdomian Decomposition method is developed to obtain an approximate solution of the nonlinear system of Fractional Differential Equations. This method combines Mahgoub transform and Adomian decomposition method. Here, the nonlinear terms have been handled with the help of Adomian polynomials. The fractional derivatives are described in the Caputo sense. Some examples are given to demonstrate that our newly method is very efficient and accurate.