Analytic and numerical solutions for systems of fractional Schrödinger equation

Groundwater flow in media with dual porosity has long been an active area of research given the complexities that arise during such modelling. In this pursuit, various approaches have been utilized to capture some variational behaviours – methods of systems of differential equations as well as the application of some fractional differential operators. However, these models fail to accurately capture the crossover behaviour and the heterogeneity of the geological formations. In the present study, the authors make use of the piecewise modelling in dual media to capture the crossover behaviour. To achieve this, they consider fractional differential and integral operators with different kernels including the power law, exponential decay, and the generalized Mittag–Leffler functions. Furthermore, randomness is incorporated in the modified model to obtain a piecewise fractional stochastic system of partial differential equations. The numerical method based on the Newton polynomial interpolation is applied to solve the obtained systems of partial differential equations numerically. MATLAB is then used to generate simulations. For a fractional order starting from 0.95 to 0.7, the results display high flow velocity within fractures; this behaviour is known as long range dependency. Also within the matrix soil, the authors observe a high flow, meaning in this case that the fractional derivative is expressing the flow within a matrix rock with high transmissivity. For a fractional order ranging from 0.7 to 0.5, the results show normal flow behaviours; water flows with average velocity within the matrix rock and moderate to high velocity within fractures. From 0.5 to 0.2, the authors observe slow flow behaviour, the mathematical equations exhibiting flow with low velocities, expressing a flow within shale with limited or no transmissivity. Keywords: dual porosity non-local operators piecewise modelling groundwater modelling transmissivity

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.

Solving systems of fractional differential equations using differential transform method

Existence of series solutions for certain nonlinear systems of time fractional partial differential equations

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.

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.

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.

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.

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.

Positive solutions of a system of non-autonomous fractional differential equations

This paper studies a system of nonlinear fractional differential equations (FDEs) with deviated arguments. Many linear and nonlinear problems are faced in the real-life. Generally, linear problems are solved quickly, but some difficulties appear while solving nonlinear problems. Our purpose is to approximate those solutions numerically via the Adomian decomposition method (ADM). Here, our main goal is to apply the ADM to solve higher-order nonlinear system of FDEs with deviated arguments. We prove the existence and uniqueness of the solution using Banach contraction principle. Moreover, we plot the figures of ADM solutions using MATLAB.

On a linear manifold of the space of square summable functions on a finite segment vanishing at its ends, we consider the operator of left-sided Caputo fractional differentiation. We prove that the adjoint for it is the operator of right-sided Caputo fractional differentiation. Similar results are established for the Riemann–Liouville fractional differentiation operators. We also demonstrate that the operator, which is represented as the sum of the left-sided and the right-sided fractional differentiation operators is self adjoint. The known properties of the Caputo and Riemann–Liouville fractional derivatives are used to substantiate the results.

A constant proportional caputo operator for modeling childhood disease epidemics

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