Fractional Euler numbers and generalized proportional fractional logistic differential equation

In this paper, we present an operational matrix method to obtain numerical solutions of 𝜓-Caputo fractional ordinary and partial differential equations. For this purpose, a fractional version of the Taylor theorem is presented in the framework of 𝜓-fractional calculus. The method converts the underlying ordinary or partial differential equations to systems of algebraic equations. The method is accompanied by numerical examples to verify the applicability and effectiveness of the proposed method. Further, estimates of upper bounds of error for the approximations have been derived.

Section: Generalized Fractional Taylor Theoremmentioning

confidence: 99%

“…For the existence and uniqueness of the mild solution of the fractional integro‐differential with the nonlocal initial condition described by the Caputo fractional operator, we refer the reader to Sene 8 and references therein. Nieto 9 solved a logistic differential equation for generalized proportional Caputo fractional derivative.…”

Section: Introductionmentioning

confidence: 99%

A generalized Taylor operational matrix method for ψ‐fractional differential equations

Tariq

Rehman

Saeed

2022

“…15 Numerous mathematicians and physicists contributed to the improvement of the hypotheses of fragmentary calculus. [16][17][18][19] Fractional calculus has been studied by a significant number of scientists and others,. [20][21][22] Recent experiments in the physical sciences and engineering have shown that a wide range of nonclassical phenomena can be modeled by calculus fractional derivatives.…”

Section: Introductionmentioning

confidence: 99%

“…Numerous mathematicians and physicists contributed to the improvement of the hypotheses of fragmentary calculus 16–19 . Fractional calculus has been studied by a significant number of scientists and others, 20–22 .…”

Section: Introductionmentioning

confidence: 99%

Theories of tempered fractional calculus applied to tempered fractional Langevin and Vasicek equations

Obeidat

Rawashdeh

2023

Our main goal in the current research work is to explore proofs of newly discovered theorems related to tempered fractional calculus. We use a new mechanism, namely, the natural tempered fractional transformation method, which can be used to solve important tempered fractional differential equations that are important in science, such as the linear and nonlinear tempered fractional differential equations. Indeed, we found new exact solutions to both tempered fractional Langevin and Vasicek differential equations and an exact solution for the famous tempered fractional diffusion equation. The new method makes it easier to do the calculations than with the traditional methods, and all you need is a few simple manipulations. Our new research technique is straightforward to use and highly accurate.

“…The logistic differential equation has many applications in different fields. Recently, the fractional version of the logistic equation has been considered by several authors [5,6,[9][10][11][12][13]. Power series representation of the solution of the fractional logistic equation and the existence of solution is discussed in Area and Nieto [10].…”

Section: Introductionmentioning

confidence: 99%

Hybrid Fibonacci wavelet method to solve fractional‐order logistic growth model

Ahmed

Jahan

Nisar

2023

The aim of this study is to develop the Fibonacci wavelet method together with the quasi‐linearization technique to solve the fractional‐order logistic growth model. The block‐pulse functions are employed to construct the operational matrices of fractional‐order integration. The fractional derivative is described in the Caputo sense. The present time‐fractional population growth model is converted into a set of nonlinear algebraic equations using the proposed generated matrices. Making use of the quasi‐linearization technique, the underlying equations are then changed to a set of linear equations. Numerical simulations are conducted to show the reliability and use of the suggested approach when contrasted with methods from the existing literature. A comparison of several numerical techniques from the available literature is presented to show the efficacy and correctness of the suggested approach.