Using Shehu integral transform to solve fractional order Caputo type initial value problems

Abstract

In the present research analysis, linear fractional order ordinary differential equations with some defined condition (s) have been solved under the Caputo differential operator having order α > 0 via the Shehu integral transform technique. In this regard, we have presented the proof of finding the Shehu transform for a classical nth order integral of a piecewise continuous with an exponential order function which leads towards devising a theorem to yield exact analytical solutions of the problems under investigation. Varying fractional types of problems are solved whose exact solutions can be compared with solutions obtained through existing transformation techniques including Laplace and Natural transforms.

New chaotic attractors: Application of fractal‐fractional differentiation and integration

“…[16][17][18][19][20][21][22][23][24][25][26] Very recently, the application of this concept was extended to capture chaotic attractors and even to describe the spread of infectious diseases within a given population. [27][28][29][30][31][32][33][34][35][36][37] The results obtained from these studies were similar to those obtained from classical fractional differential and integral operators, a proof and validity of nonlocality of this calculus more precisely the associate integral display properties similar to those of the Riemann-Liouville integral. While, in the literature, there are some published articles where these differential operators are discredited even, in this article, it is claimed that they are not differential operators; the results obtained from models show that the differential operators appear in many situations in nature.…”

“…Some more applications of this concept can be found in diffusion, groundwater flow problem, and groundwater pollution problems 16–26 . Very recently, the application of this concept was extended to capture chaotic attractors and even to describe the spread of infectious diseases within a given population 27–37 . The results obtained from these studies were similar to those obtained from classical fractional differential and integral operators, a proof and validity of nonlocality of this calculus more precisely the associate integral display properties similar to those of the Riemann‐Liouville integral.…”

“…. Figure 18A-F shows numerical simulations for the nine-term 3D novel chaotic system with four quadratic nonlinearities in Atangana-Baleanu-Caputo sense (35) considering different values of and , arbitrarily chosen.…”

Homotopy perturbation Shehu transform method for solving fractional models arising in applied sciences

“…To mention a few, we have the homotopy perturbation method (HPM) [6], the Adomian decomposition method (ADM) [7], the Laplace decomposition method (LDM) [8], the homotopy perturbation transform method (HPTM) [9], and so on. Besides using the Laplace-type integral transform [10,11], some new efficient iterative techniques with the Caputo fractional derivative [12] and Atangana-Baleanu fractional derivative [13] are developed, for example, see [14][15][16][17][18][19][20][21][22][23][24][25]. Those iterative algorithms are successfully applied to many applications in applied physical science.…”

Fractional optimal control problems on a star graph: Optimality system and numerical solution

“…Fractional order dynamical systems appear in several works, especially in viscoelasticity and in hereditary solid mechanics. Recently, various practical applications of the fractional calculus using Caputo fractional derivative has been addressed by Qureshi et al in [6,33,34,35,36]. Considerable work has been done in the area of fractional optimal control problem on the real line.…”