Fractional Differential Equations Involving the Spherical Bessel Function j0: Analytical Solutions Via Laplace Transform | Chapter 03 | Physical Science: New Insights and Developments Vol. 4

 

This work addresses the resolution of fractional differential equations whose nonhomogeneous part is given by the spherical Bessel function 𝐽₀(𝑥). By using the fractional derivative in the sense of Caputo and the Laplace transform, a general analytical solution is obtained in terms of the generalised hypergeometric functions 2^𝐹3, revealing a recurrent structure in the solutions. Furthermore, particular cases for integer and fractional orders are presented, highlighting the appearance of special functions such as the sine integral and Fresnel functions. The results confirm the close relationship between fractional calculus and Bessel functions, proposing new perspectives for applications in mathematical physics.

 

Author(s) Details :-

 

Jorge Olivares Funes
Department of Mathematics, Universidad de Antofagasta, Antofagasta, Chile.

 

Pablo Martin
Department of Physics, Universidad de Antofagasta, Antofagasta, Chile.

 

Elvis Valero Kari
Universidad Mayor de San Andres, Bolivia.

 

Maria Teresa Veliz Aviles
Department of Physics, Universidad de Antofagasta, Antofagasta, Chile.

 

Please see the book here :- https://doi.org/10.9734/bpi/psniad/v4/6785

The Multistage Homotopy-Perturbation Method: A Powerful Scheme for Handling a Fractional Lorenz System | Chapter 12 | Research Highlights in Science and Technology Vol. 5

 In this division, the multistage homotopy perturbation method (MHPM) is used to present analytical resolutions of the time fractional Lorenz order in Caputo sense. This method is a qualification of the standard homotopy perturbation method (HPM) that yields an analytical solution in conditions of a convergent limitless power series accompanying easily calculable terms. Some numerical correspondings between the (MHPM) and (HPM) with the 4th order Runge-Kutta pattern (RK4) is presented. Moreover, we strengthen our results by constructing the residual mistake for the solution. It is decided that the (MHPM) is reliable and effective form in finding analytical answers and it is very easy to build an accurate approximate solution for the partial Lorenz system.

Author(s) Details:

M. A. Abu-Ibrahim,
Mathematics Department, Faculty of Science, Tanta University, Tanta, Egypt.

Please see the link here: https://stm.bookpi.org/RHST-V5/article/view/11165

Study on the Application of the Fractional Calculus in Pharmacokinetic Modeling | Chapter 03 | Technological Innovation in Engineering Research Vol. 2

 The application of fractional calculus (FC) in biomedicine is discussed in this paper. We present three different integer order pharmacokinetic models with two and three compartments that are commonly used in cancer therapy, and we solve them numerically and analytically to show the absorption, distribution, metabolism, and excretion (ADME) of drugs or nanoparticles (NPs) in various tissues. Because tumour cell interactions are systems with memory, the fractional order framework, rather than ordinary and delay differential equations, is a superior way to explain cancer processes. To discretize the model and produce the fractional order form to represent the fractal processes of drug transport in the body, the nonstandard finite difference analysis or NSFD approach following the Grunwald-Letinkov discretization can be used. Implementing a simple and efficient numerical approach to solve these fractional order models will be critical. In order to develop the numerical solution of fractional order two and tri-compartmental pharmacokinetic models for oral drug administration, numerical techniques based on the finite difference scheme were used. The fractional order model extends the capabilities of the integer order model into the extended domain of fractional calculus, as demonstrated in this work. Furthermore, fractional order modelling gives more power to control the dynamical behaviours of the (ADME) process in different tissues because the order of fractional derivative can be used as a new control parameter to extract a variety of governing classes on the nonlocal behaviours of a model; however, integer order modelling gives more power to control the dynamical behaviours of the (ADME) process in different tissues. Only the local and integer order domains are addressed by the order operator. In reality, for this sort of application, NSFD may be employed as an effective and simple way to implement, and it provides a handy framework for solving the proposed fractional order models.

Author(S) Details

Tahmineh Aziz
Florida State University, US.

View Book:- https://stm.bookpi.org/TIER-V2/article/view/6839

Neuronal Spiking Patterns of Fractional-order Morris-Lecar Model | Chapter 01 | Technological Innovation in Engineering Research Vol. 2

 Fractional calculus is a novel method for studying biological and physical processes that include memory effects. Fractional calculus examines the nonlinear behaviour of physical and biological systems with varying degrees of fractionality or fractality using differential and integral operators, including non-integer orders. Because the fractional derivative can better describe the long memory features of neural responses, we expand the integer-order Morris-Lecar model in the fractional-order domain to better characterise neuron dynamics in this work. The fractional calculus was used to create this novel mathematical model to examine the complicated spiking patterns of fractional-order Morris-Lecar neural systems. We compare the findings to the Morris-Lecar model with integer order. These equations do not have analytical solutions that may be found explicitly. As a result, in order to discover the dynamical characteristics of solutions. We employed numerical and approximation approaches. Morris-Lecar reproduces quiescent, spiking, and bursting activities in the same way as its original model, but with a greater input current, depending on the different parameter values for the fractional-order Morris-Lecar. For varied input current and derivative orders, we numerically uncover the hopf bifurcation, saddle node bifurcation of limit cycle, and homocinic bifurcation for this model. Using the benefits of the fractional order derivative, we establish distinct classes of this model for a range of orders, allowing us to better extract all of the intricate dynamics of this single neuron model.

Author(S) Details

Tahmineh Azizi
Department of Mechanical Engineering, Florida State University, Florida, USA.

Robert Mugabii
College of Veterinary Medicine, Iowa State University, Iowa, USA.

View Book:- https://stm.bookpi.org/TIER-V2/article/view/6837