Search Results (13)

We analyze the error estimates of a fully discrete scheme for solving a semilinear stochastic subdiffusion problem driven by integrated fractional Gaussian noise with a Hurst parameter $H\in (0,1)$. The covariance operator Q of the stochastic fractional Wiener process satisfies $\parallel A^{-\rho }{Q}^{1/2}{\parallel }_{HS}<\infty$ for some $\rho \in [0,1)$, where ${\parallel ·\parallel }_{HS}$ denotes the Hilbert–Schmidt norm. The Caputo fractional derivative and Riemann–Liouville fractional integral are approximated using Lubich’s convolution quadrature formulas, while the noise is discretized via the Euler method. For the spatial derivative, we use the spectral Galerkin method. The approximate solution of the fully discrete scheme is represented as a convolution between a piecewise constant function and the inverse Laplace transform of a resolvent-related function. By using this convolution-based representation and applying the Burkholder–Davis–Gundy inequality for fractional Gaussian noise, we derive the optimal convergence rates for the proposed fully discrete scheme. Numerical experiments confirm that the computed results are consistent with the theoretical findings. Full article

Inspired by prior research on fractional calculus, we introduce new fractional integral and derivative operators: the $({\rho }_1,{\rho }_2,k_1,k_2,\phi )$-proportional integral and the $({\rho }_1,{\rho }_2,k_1,k_2,\phi )$-proportional Hilfer fractional derivative. Numerous previous studied fractional integrals and derivatives can be considered as particular instances of the novel operators introduced above. Some properties of the $({\rho }_1,{\rho }_2,k_1,k_2,\phi )$-proportional integral are discussed, including mapping properties, the generalized Laplace transform of the $({\rho }_1,{\rho }_2,k_1,k_2,\phi )$-proportional integral and $({\rho }_1,{\rho }_2,k_1,k_2,\phi )$-proportional Hilfer fractional derivative. The results obtained suggest that the most comprehensive formulation of this fractional calculus has been achieved. Under the guidance of the findings from earlier sections, we investigate the existence of mild solutions for the $({\rho }_1,{\rho }_2,k_1,k_2,\phi )$-proportional Hilfer fractional Cauchy problem. An illustrative example is provided to demonstrate the main results. Full article

In the presented research, the uniqueness and existence of a mild solution for a fractional system of semilinear evolution equations with infinite delay and an infinitesimal generator operator are demonstrated. The generalized Liouville–Caputo derivative of non-integer-order $1<\alpha \le 2$ and the parameter $0<\rho <1$ are used to establish our model. The $\rho$-Laplace transform and strongly continuous cosine and sine families of uniformly bounded linear operators are adapted to obtain the mild solution. The Leray–Schauder alternative theorem and Banach contraction principle are used to demonstrate the mild solution’s existence and uniqueness in abstract phase space. The results are applied to the fractional wave equation. Full article

In this article, the $\rho$-Laplace transform is paired with a new iterative method to create a new hybrid methodology known as the new iterative transform method (NITM). This method is applied to analyse fractional-order third-order dispersive partial differential equations. The suggested technique procedure is straightforward and appealing, and it may be used to solve non-linear fractional-order partial differential equations effectively. The Caputo operator is used to express the fractional derivatives. Four numerical problems involving fractional-order third-order dispersive partial differential equations are presented with their analytical solutions. The graphs determined that their findings are in excellent agreement with the precise answers to the targeted issues. The solution to the problems at various fractional orders is achieved and found to be correct while comparing the exact solutions at integer-order problems. Although both problems are the non-linear fractional system of partial differential equations, the present technique provides its solution sophisticatedly. Including both integer and fractional order issues, solution graphs are carefully drawn. The fact that the issues’ physical dynamics completely support the solutions at both fractional and integer orders is significant. Moreover, despite using very few terms of the series solution attained by the present technique, higher accuracy is observed. In light of the various and authentic features, it can be customized to solve different fractional-order non-linear systems in nature. Full article

In this article, we study the hitting probability of a circumference $C_R$ for a correlated Brownian motion $\underline{B}\left(t\right)=\left(B_1\left(t\right),B_2\left(t\right)\right)$, $\rho$ being the correlation coefficient. The analysis starts by first mapping the circle $C_R$ into an ellipse E with semiaxes depending on $\rho$ and transforming the differential operator governing the hitting distribution into the classical Laplace operator. By means of two different approaches (one obtained by applying elliptic coordinates) we obtain the desired distribution as a series of Poisson kernels. Full article

Positive linear electrical circuits systems described by generalized fractional derivatives are studied in this paper. We mainly focus on the reachability and observability of linear electrical circuits systems. Firstly, generalized fractional derivatives and $\rho$-Laplace transform of f is presented and some preliminary results are provided. Secondly, the positivity of linear electrical circuits systems described by generalized fractional derivatives is investigated and conditions for checking positivity of the systems are derived. Thirdly, reachability and observability of the generalized fractional derivatives systems are studied, in which the $\rho$-Laplace transform of a Mittag-Leffler function plays an important role. At the end of the paper, illustrative electrical circuits systems are presented, and conclusions of the paper are presented. Full article

In this article, we use the homotopy perturbation transform method to find the fractional Kersten–Krasil’shchik coupled Korteweg–de Vries (KdV) non-linear system. This coupled non-linear system is typically used to describe electric circuits, traffic flow, shallow water waves, elastic media, electrodynamics, etc. The homotopy perturbation method is modified with the help of the $\rho$-Laplace transformation to investigate the solution of the given examples to show the accuracy of the current technique. The solution of the given technique and the actual results are shown and analyzed with figures. Full article

In this article, we also introduced two well-known computational techniques for solving the time-fractional Fornberg–Whitham equations. The methods suggested are the modified form of the variational iteration and Adomian decomposition techniques by $\rho$-Laplace. Furthermore, an illustrative scheme is introduced to verify the accuracy of the available methods. The graphical representation of the exact and derived results is presented to show the suggested approaches reliability. The comparative solution analysis via graphs also represented the higher reliability and accuracy of the current techniques. Full article

This paper addresses the solution of the incompressible second-grade fluid models. Fundamental qualitative properties of the solution are primarily studied for proving the adequacy of the physical interpretations of the proposed model. We use the Liouville-Caputo fractional derivative with its generalized version that gives more comprehensive physical results in the analysis and investigations. In this work, both the $\rho$-Laplace homotopy transform method ($\rho$-LHTM) and the heat balance integral method (HBIM) are successfully combined to solve the fractional incompressible second-grade fluid differential equations. Numerical simulations and their physical interpretations of the mentioned incompressible second-grade fluid model are ensured to illustrate the main findings. It is also proposed that one can recognize the differences in physical analysis of diffusions such as ballistic diffusion, super diffusion, and subdiffusion cases by considering the impact of the orders $\rho$ and $\phi$. Full article

The aim of this paper is to study the stability of generalized Liouville–Caputo fractional differential equations in Hyers–Ulam sense. We show that three types of the generalized linear Liouville–Caputo fractional differential equations are Hyers–Ulam stable by a $\rho$ -Laplace transform method. We establish existence and uniqueness of solutions to the Cauchy problem for the corresponding nonlinear equations with the help of fixed point theorems. Full article

The behavior of the analytical solutions of the fractional differential equation described by the fractional order derivative operators is the main subject in many stability problems. In this paper, we present a new stability notion of the fractional differential equations with exogenous input. Motivated by the success of the applications of the Mittag-Leffler functions in many areas of science and engineering, we present our work here. Applications of Mittag-Leffler functions in certain areas of physical and applied sciences are also very common. During the last two decades, this class of functions has come into prominence after about nine decades of its discovery by a Swedish Mathematician Mittag-Leffler, due to the vast potential of its applications in solving the problems of physical, biological, engineering, and earth sciences, to name just a few. Moreover, we propose the generalized Mittag-Leffler input stability conditions. The left generalized fractional differential equation has been used to help create this new notion. We investigate in depth here the Lyapunov characterizations of the generalized Mittag-Leffler input stability of the fractional differential equation with input. Full article

In this paper, the approximate solutions of the fractional diffusion equations described by the fractional derivative operator were investigated. The homotopy perturbation Laplace transform method of getting the approximate solution was proposed. The Caputo generalized fractional derivative was used. The effects of the orders $\alpha$ and $\rho$ in the diffusion processes was addressed. The graphical representations of the approximate solutions of the fractional diffusion equation and the fractional diffusion-reaction equation both described by the Caputo generalized fractional derivative were provided. Full article

In this paper the process of semi-Markovian random walk with negative drift under angle α (0° < α < 90°), and positive jumps with probability ρ (0 < ρ < 1) having two delaying screens at level zero and a (a > 0) is constructed. The exact expressions for Laplace transforms of the distributions of the first moments in order to reach to these screens by the process and, in particularly, the expectations and the variances of indicated distributions are obtained. Full article