A new method for one- and higher-dimensional linear and nonlinear tempered Caputo fractional diffusion-type equations

In this paper we study linear and nonlinear fractional diffusion equations with the Caputo fractional derivative of non-singular kernel that has been launched recently (Caputo and Fabrizio in Prog. Fract. Differ. Appl. 1(2):73-85, 2015). We first derive simple and strong maximum principles for the linear fractional equation. We then implement these principles to establish uniqueness and stability results for the linear and nonlinear fractional diffusion problems and to obtain a norm estimate of the solution. In contrast with the previous results of the fractional diffusion equations, the obtained maximum principles are analogous to the ones with the Caputo fractional derivative; however, extra necessary conditions for the existence of a solution of the linear and nonlinear fractional diffusion models are imposed. These conditions affect the norm estimate of the solution as well.

In this paper, we examined a wide class of the variable order fractional problems such as linear and nonlin-ear fractional variable order differential equations, variable order fractional functional boundary value problems, variable order fractional pantograph differential equations. The proposed method is a collocation method based on the Bessel polynomials and the operational matrix of derivatives, which transformed equations into a system of non-linear algebraic equations to achieve the approximate solution. By using Caputo fractional derivative, the operational matrix of the variable-order fractional derivatives is constructed. The error analysis shows that the method is convergent. Several numerical results confirm the accuracy and efficiency of the proposed method. Keywords: Bessel collocation method, Variable-order fractional operational matrix, Variable order fractional differential equations, Variable order fractional functional boundary value problems, Variable order fractional panto-graph equations.

In this paper, by using the collocation method the operational matrix of Caputo fractional derivative was constructed for the cubic B-spline scaling function and wavelets. We apply the operational matrix and the Collocation method to solve linear fractional Volterra integro-differential equations. By using the principal characteristic of this technique, we have the operational matrix, which is applied to solve any linear fractional Volterra integro-differential equation by reducing the time, computer memory occupation and convert to a system of linear equations. Finally, the validity and applicability of the new technique will be shown by some numerical examples and convergence analysis.

Application of orthonormal Bernstein polynomials to construct a efficient scheme for solving fractional stochastic integro-differential equation

This chapter presents basic definitions and characteristics of fractional calculus. Definitions and characteristics of the Riemann–Liouville fractional integral and derivative, Caputo fractional derivative, Grünwald–Letnikov fractional derivative, Riesz fractional derivative, modified Riemann–Liouville derivative, and local fractional derivative have been primarily explored here. However, the Riemann–Liouville definitions of fractional differentiation play a significant role in developing fractional calculus. Because the derivative of constant is not zero, the Riemann–Liouville definition is not widely applicable. In such scenarios, the Caputo fractional derivative is more suited than the standard Riemann–Liouville derivative for applications in various engineering problems. It may be noted that the Riemann–Liouville technique is challenging to understand physically, while the Caputo approach employs integer-order initial conditions, which are easier to apply in real-world applications. This chapter also covers the Euler psi function, incomplete gamma function, beta function, incomplete beta function, Wright function, Mellin-Ross function, error function, hypergeometric functions such as Gauss, Kummer, and extended hypergeometric functions, and the H -function.

In this paper, we analyse the propagation of a small density perturbation in a one-dimensional compressible fluid by means of fractional calculus modelling, replacing thus the ordinary time derivative with the Caputo fractional derivative in the constitutive equations. By doing so, we embrace a vast phenomenology, including subdiffusive, superdiffusive, and also memoryless processes such as classical diffusions. From a mathematical point of view, we study systems of coupled fractional equations, leading to fractional diffusion equations or to equations with sequential fractional derivatives. In this framework, we also propose a method to solve partial differential equations with sequential fractional derivatives by analysing the corresponding coupled system of equations.

In this study, a wavelet method is developed to solve a system of nonlinear variable-order (V-O) fractional integral equations using the Chebyshev wavelets (CWs) and the Galerkin method. For this purpose, we derive a V-O fractional integration operational matrix (OM) for CWs and use it in our method. In the established scheme, we approximate the unknown functions by CWs with unknown coefficients and reduce the problem to an algebraic system. In this way, we simplify the computation of nonlinear terms by obtaining some new results for CWs. Finally, we demonstrate the applicability of the presented algorithm by solving a few numerical examples.

This paper presents a linearized finite difference scheme for solving a kind of time-space fractional nonlinear diffusion-wave equations with initial singularity, where the Caputo fractional derivative in time and the Riesz fractional derivative in space are involved. First, the considered problem is equivalently transformed into its partial integro-differential form. Then, the fully discrete scheme is constructed by using the Crank–Nicolson technique, the L1 approximation, and the convolution quadrature formula to deal with the temporal discretizations. Meanwhile, the classical central difference formula and the fractional central difference formula are applied to approximate the second-order derivative and the Riesz fractional derivative in space, respectively. Moreover, the stability and convergence of the proposed scheme are strictly proved by using the discrete energy method. Finally, some numerical experiments are presented to illustrate the theoretical results.

In this article, the fractional derivatives in the sense of modified Riemann–Liouville and the exp-function method are used to construct exact solutions for some nonlinear partial fractional differential equations via the nonlinear fractional Liouville equation and nonlinear fractional Zoomeron equation. These nonlinear fractional equations can be turned into another nonlinear ordinary differential equation by complex transform method. This method is efficient and powerful in solving wide classes of nonlinear fractional order equations. The exp-function method appears to be easier and more convenient by means of a symbolic computation system.

The article aims to propose the Lerch operational matrix method to solve a stochastic fractional differential equation. In this approach, the Lerch polynomials have been used as a basis function. Then, the product operational matrix, integral operational matrix, stochastic operational matrix, and operational matrix of fractional integral based on the Lerch polynomials have been constructed. The main characteristic of this method is to reduce the stochastic fractional differential equation into a system of algebraic equations by using derived operational matrices and suitable collocation points. Moreover, the convergence and error analysis of the presented method is also discussed in detail. Additionally, the applicability of the proposed technique is also demonstrated by solving some examples. To confirm the accuracy and effectiveness of the suggested technique, a comparison between the results produced by the proposed method and those obtained by other methods has been provided.

This work presents an alternative solution for the mathematical analysis of the fractional waves in dielectric media. For the fractional wave equation, the Caputo fractional derivative was considered, the order of the spatial and temporal fractional derivatives are , respectively. In this analysis, we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space and time derivatives into the fractional wave equation. We will consider source free Maxwell equations in isotropic and homogeneous dielectric medium. The general solutions obtained in our research have been expressed in terms of the multivariate Mittag–Leffler functions, these functions depend only on the parameters and preserving the appropriated physical units according to the system studied.

Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation

This research paper is about the existence and uniqueness of the coupled system of nonlinear fractional Langevin equations with multipoint and nonlocal integral boundary conditions. The Caputo fractional derivative is used to formulate the fractional differential equations, and the fractional integrals mentioned in the boundary conditions are due to Atangana–Baleanu and Katugampola. The existence of solution has been proven by two main fixed-point theorems: O’Regan’s fixed-point theorem and Krasnoselskii’s fixed-point theorem. By applying Banach’s fixed-point theorem, we proved the uniqueness result for the concerned problem. This research paper highlights the examples related with theorems that have already been proven.

This study aims to propose novel Zernike wavelets and a new method based on the operational matrices for solving nonlinear fractional differential equations. First, non-orthogonal Zernike wavelets are introduced using the Zernike polynomials. Then, a new technique based on combining these wavelets with the block pulse functions is presented to derive the operational matrix of fractional integration and to solve nonlinear fractional differential equations. Moreover, an error analysis is conducted by providing required theorems. Besides, the proposed method is employed to solve a nonlinear fractional competition model of breast cancer. Finally, a parametric study is performed to consider the effect of fractional order on the population of healthy, cancer stem, tumour, and immune cells, as well as the excess estrogen.

The generalized block-pulse operational matrices are derived as integral operators for operational calculus. In comparison with Walsh tables, the generalized operational matrices are nothing but the block-pulse tables. Further, it is pointed out that the conventional block-pulse operational matrix is a special case of the generalized operational matrices. Also, the generalized operational matrices are preferable to conventional block-pulse operational matrix when a given function is integrated repeatedly. Finally, the inverse Laplace transform of a rational transfer function via the generalized operational matrices is illustrated as an application of operational calculus.