Fox H-functions as exact solutions for Caputo type mass spring damper system under Sumudu transform

Abstract

Closed form solutions for mathematical systems are not easy to find in many cases. In particular, linear systems such as the population growth/decay model, RLC circuit, mixing problems in chemistry, first-order kinetic reactions, and mass spring damper system in mechanical and mechatronic engineering can be handled with tools available in theoretical study of linear systems. One such linear system has been investigated in the present research study. The second order linear ordinary differential equation called the mass spring damper system is explored under the Caputo type differential operator while using the Sumudu integral transform. The closed form solution has been found in terms of the Fox H-function wherein different aspects of the solution can be obtained with variation in α ∈ (1, 2] and β ∈ (0, 1]. The classical mass spring damper model is retrieved for α = β = 1.

Rothe's method for solving multi‐term Caputo–Katugampola fractional delay integral diffusion equations

“…It is remarkable that Example 2 comes from partial population model, that is, partial population equation with diffusion and delay. In particular, we note that, very recently, Baleanu et al and Qureshi et al 1,2,4,[6][7][8][9][10][11] considered some dynamic behaviors for different kinds of fractional equations. As a future research plan, we aim to study some multi-term fractional dynamical equation problems.…”

“…) is reduced to (3) of Migórski and Zeng. 16 It is well known that the fractional derivative in sense of Caputo-Katugampola includes the Caputo fractional derivatives in previous studies, 4,[7][8][9][10][11] which is the first novelty of this paper. Moreover, Equation (1.1) contains multi-term calculus, which is different from single-term derivative in previous studies, 7,8,11,15,16 which is the second novelty of this paper.…”

Comparative analysis of Riemann-Liouville, Caputo-Fabrizio, and Atangana-Baleanu integrals

“…This scheme is presented in a simplified form to numerically evaluate the CF integral of y(t). It may be noted that when α = 1 and M(α) from ( 16) in CF integral (14), one retrieves the ordinary integral from the classical calculus. In addition to this, the upper bound for left rectangular rule from classical quadrature is found to be…”

“…Fractional calculus (differentiation and integration of order α ∈ R + ), earlier considered to be a branch of pure mathematical analysis, has now been found to be extremely useful in various fields of applied sciences. It plays a vital role in modeling diverse physical and natural phenomena including neural networks [1], bio-electrodes [2], bio-materials [3], mathematical biology and bifurcations [4][5][6][7], finance and economics [8], electrical and mechanical engineering [9][10][11][12][13][14][15], fluid dynamics [16], control systems [17], plant genetics [18][19][20] and many more. One of the major reasons for the rapidly increasing popularity of this field is its capability of modeling all those dynamic systems which have history (memory) effects and anomalous behavior, which is something common in most of the physical and natural systems.…”

Numerical solution of two-dimensional Fredholm integro-differential equations by Chebyshev integral operational matrix method

“…For instance, [14] used a shifted 1D Chebyshev polynomial for the solution of 3D Volterra integral equations of the second kind. Various mathematical models, such as [15][16][17][18][19][20] and most of the references referenced therein, have been redefined in the context of fractional calculus and in epidemiology. In other to improve on the existing methods in the literature, this paper therefore presents a Chebyshev integral operational matrix method (CIOMM) for the numeri-cal solutions of 2D Fredholm integro-differential equations.…”