Existence Results for Resonant Functional Boundary Value Problems with Generalized Weighted Fractional Derivatives

In this study, the peridynamic Drucker‐Prager plastic model with fractional order derivative is proposed to investigate the plastic behavior of surrounding rocks around tunnels, in which the Caputo fractional derivative is employed due to its mathematical simplicity. Instead of utilizing just one parameter for the typical nonassociated flow rule, such as dilation angle, multiple parameters, such as fractional order and interval of fractional derivative, are used to specify the direction of plastic deformation, and the fractional derivative based‐typical flow rule is proposed. As a result, compared with the traditional peridynamic Drucker‐Prager plastic model, the proposed method is a more generalized model including the typical nonassociated flow rule. Besides, based on the PD force density, the strategy of exertion of initial stress is proposed to simulate in‐situ stress in rocks. Taking a block subjected to compression as an example, the impacts of various factors, such as fractional order and interval of fractional derivative, are investigated. A numerical simulation of tunnel excavation in rocks is carried out and the numerical results obtained by the proposed method are verified by comparing with FEM results.

<abstract><p>The aim of this paper is to demonstrate a coupled system of second-order fractional pantograph differential equations with coupled four-point boundary conditions. The proposed system involves Atangana-Baleanu-Caputo ($ \mathcal{ABC} $) fractional order derivatives. We prove the solution formula for the corresponding linear version of the given system and then convert the system to a fixed point system. The existence and uniqueness results are obtained by making use of nonlinear alternatives of Leray-Schauder fixed point theorem, and Banach's contraction mapping. In addition, the guarantee of solutions for the system at hand is shown by the stability of Ulam-Hyers. Pertinent examples are provided to illustrate the theoretical results.</p></abstract>

Fractional order entropy: New perspectives

The purpose of this research is to establish conditions that guarantee the solvability of a resonant third-order boundary value problem (BVP) on the half-line. The resonant problem is subject to integral and m-point boundary conditions while, the kernel of the differential operator is equal to two. The coincidence degree theory will be applied and an illustrative example will be considered to demonstrate the result.

In this article, the authors report the Chebyshev pseudospectral method for solving twodimensional nonlinear Schrodinger equation with fractional order derivative in time and space both. The modified Riemann-Liouville fractional derivatives are used to define the new fractional derivatives matrix at CGL points. Using the Chebyshev fractional derivatives matrices, the given problem is reduced to a diagonally block system of nonlinear algebraic equations, which will be solved using Newton's Raphson method. The proposed methods have shown error analysis without any dependency on time and space step restrictions. Some model examples of the equations, defined on a convex and rectangular domain, have tested with various values of fractional order <i>α</i> and <i>β</i>. Moreover, numerical solutions are demonstrated to justify the theoretical results.

Fractional derivative approach to non-Darcian flow in porous media

A comparison of a fractional derivative model with an empirical model for non-linear shock waves in swelling shales

General Fractional Calculus Operators with the Sonin kernels and Some of Their Applications

Analysis of perturbation factors and fractional order derivatives for the novel singular model using the fractional Meyer wavelet neural networks

Nanofluids are gaining extensive attention due to their thermo‐physical properties in technological and industrial fields for controlling the effects of heat transfer. Classical nanofluid studies are generally confined to models described by partial differential equations of an integer order where the memory effect and hereditary properties of materials are neglected. In order to overcome these downsides, the present work focuses on studying nanofluids with fractional derivative formed by differential equations with Caputo time derivatives that provides memory effect on nanofluid characteristics. Also, investigation on natural convective flow, heat, and mass transfer of nanofluids formed by different base fluids with different shapes of copper nanoparticles past an infinite vertical plate with radiation effect is carried out. The governing fractional differential equations are solved by employing Laplace transform technique with suitable boundary conditions. The different base fluids—water (H2O), SA:sodium alginate (C6H9Na O7), and EG:ethylene glycol (C2H6O2) and different shapes of nanoparticles—blade, brick, platelet, and cylinder are considered for the study. The exact solutions are obtained for the temperature, concentration, and velocity distributions and the respective Nusselt number, Sherwood number, and skin‐friction coefficient. The influence of non‐dimensional parameters provides physical interpretations of temperature, concentration, and velocity fields, Nusselt number, Sherwood number, and skin friction in detail with the help of graphical representations. From the results, it is found that nanofluid with water‐based blade‐shaped nanoparticle exhibits more velocity and temperature distributions. Also, strengthen of fluid flow, temperature, and concentration of nanofluids are inversely correlate with fractional order derivatives.

At present, the theory of fractional calculus is widely used in many fields of science for modeling various processes. Differential equations with fractional derivatives are used to model the migration of pollutants in porous inhomogeneous media and allow a more correct description of the behavior of pollutants at large distances from the source. The analytical solution of differential equations with fractional order derivatives is often very complicated or even impossible. There has been proposed a numerical method for solving fractional differential equations in partial derivatives with respect to time to describe the migration of pollutants in groundwater. An implicit difference scheme is developed for the numerical solution of a non-stationary fractional differential equation, which is an analogue of the well-known implicit Crank-Nicholson difference scheme. The system of difference equations is presented in matrix form. The solution of the problem is reduced to the multiple solution of a tridiagonal system of linear algebraic equations by the tridiagonal matrix algorithm. The results of evaluating the spread of pollutant in groundwater based on the numerical method for model examples are presented. The concentrations of the substance obtained on the basis of the analytical and numerical solutions of the unsteady one-dimensional fractional differential equation are compared. The results obtained using the proposed method and on the basis of the well-known analytical solution of the fractional differential equation are in fairly good agreement with each other. The relative error is on average 9%. In contrast to the well-known analytical solution, the developed numerical method can be used to model the spread of pollutants in groundwater, taking into account their biodegradation.

Nonlinear vibration of viscoelastic beams described using fractional order derivatives

This paper is concerned with the development of a high order numerical technique for solving time‐fractional reaction–diffusion equation. The fractional derivative in the governing equation is described in the Caputo sense and a collocation method based on quintic B‐spline basis function is used to discretize the space variable. The stability and convergence analysis of the method are investigated, and it is shown that the proposed method converges to the exact solution of the problem with order of convergence O(Δt2 − α + Δx4), where α is the order of fractional derivative (0 &lt; α &lt; 1). Three test problems are considered to illustrate the efficiency and accuracy of present numerical scheme and to verify the theoretical result. It is shown that the order of fractional derivative has a profound effect on the solution of time‐fractional reaction–diffusion equation.

The Dirichlet resonant boundary value problems are considered. If the respective nonlinear equation can be reduced to a quasilinear one with a nonresonant linear part and both equations are equivalent in some domainΩand if solutions of the quasilinear problem are inΩ, then the original problem has a solution. We say then that the original problem allows for quasilinearization. If quasilinearization is possible for essentially different linear parts, then the original problem has multiple solutions. We give conditions for Emden-Fowler type resonant boundary value problem solvability and consider examples.

A fractional model of magnetohydrodynamics Oldroyd-B fluid with couple stresses, heat and mass transfer: A comparison among Non-Newtonian fluid models