A Study on Hypergeometric Solutions to Nonhomogeneous Equations of Fractional Order | Chapter 6 | Physical Science: New Insights and Developments Vol. 3

 

The study addresses the solution of a nonhomogeneous linear differential equation of fractional order α, equal to the modified Bessel function of order zero (I₀(x)), under the initial condition (f(0) = 0) and with (0 < α < 1). The Caputo definition of the fractional derivative is adopted, which is widely used in the analysis of physical and chemical phenomena such as viscoelasticity, anomalous diffusion, and electrical circuits. By means of the Laplace transform and its inverse, analytical solutions are obtained for the specific cases (α = 1/4, 1/2, 3/4), expressed in terms of hypergeometric functions 2F1. These solutions combine a fractional power of the independent variable (x) with a special function, reflecting the non-integer nature of the differential operator. Furthermore, regular patterns are observed in the parameters of the hypergeometric functions as α varies, suggesting a generalizable structure for other fractional values. The work demonstrates that fractional differential equations can be systematically solved using classical tools of integral and transform calculus, connecting fractional derivatives with families of special functions.

 

 

Author(s) Details

Jorge Olivares
University of Antofagasta, Chile.

 

Pablo Martin
University of Antofagasta, Chile.

 

Fernando Maass
University of Antofagasta, Chile.

 

Please see the book here :- https://doi.org/10.9734/bpi/psniad/v3/6604

Structuring Constitutive Equations of few common Physical Laws Vis-à-vis Types of Impulse Response Functions /Memory Kernels | Chapter 2 | Current Perspective to Physical Science Research Vol. 1

 We will study the composition of constituent expressions of various tangible laws accompanying impulse response function or thought kernel h(t) as plain bordering a river function, singular power-regulation decay function, non-singular capacity law decay function, Mittag-Leffler function, pure epidemic function and stretched epidemic function. The motivation to have this chapter search out discuss, issues about using unique and non-singular functions as fundamental impulse response function or thought kernels; in basic progress equation in several process action-and its suggestions to obtain constituent equations for differing systems. This gives a inference of system studies. We will restrict our reasoning to simple constitutive equatings of few common tangible laws that we deal in common studies. We will study two types of system accompanying evolution equation delineated as convolution that is y(t) = h(t)*x(t) (from Causality Principle). The physical laws that we argue with miscellaneous types of memory kernel are capacitor or dielectric entertainment/force/stress strain rate equations, population growth/atomic reaction equations, spread equations and wave equations. First seeing the cause (input) x(t) is equivalent to rate of change of some other physical load, i.e. x(t) ∝ f(1) (t) and second is a plan where output reaction i.e. y(t) is proportional to rate of change of cause (recommendation) x(t) i.e. y(t) ∝ x(1) (t). We note the first type of structure is like ‘response current’ to a change in applied physical ability observed in dielectric relaxations and capacitor or force to rate of change of impetus or stress and strain rate equations.  The second one complements to population growth or atomic reaction type system.  The corollary to second type of method we study where cause X is discharged by L [x] ; where L denotes a ‘Linear controller’ and y(t) ∝ x(1) (t). With L as Laplacian operator L ≡ ∇ 2, we see that we will generate various types of spread equations.  Extending further making effect as y(t) ∝ x(2) (t), we will write miscellaneous types of wave-equations. We will obtain formation of these basic essential equations accompanying zero-memory case place the memory-essence or impulse response function is a plain bordering a river-function and memorized relaxation cases accompanying singular and non-unique memory kernels or impulse reaction functions that decays with occasion. These decaying functions used as thought kernel gives a existence in which memory fades as occasion grows. However, the question stands is the memory kernel be of unique or non-singular function? We will see that for the nothing-memory case place the memory kernel is a opening function (singular in type) returns classical constitutive equatings for system that we see and use in every day physics but with the case place memory essence is other than delta function we catch constitutive equations accompanying fractional products and fractional integrations, different from what we see classically. We will note that singular function that we use beyond any doubt-decaying memory seed gives rise to combination to classical constitutive equatings where allure fractional counterpart replaces number-order (classical) derivative or integral movement. We will see that non-unique memory kernel gives be even with more complicated essential equations as weighted limitless series total of repeated integrations or weighted succession sum of partial integrations.

Author(s) Details:

Shantanu Das,
Bhabha Atomic Research Center, Mumbai, India and Department of Physics, Jadavpur University, Kolkata, India.

Please see the link here: https://stm.bookpi.org/CPPSR-V1/article/view/11916

Fractional Derivative Order with Respect to Time for Diffusion Equation: Scientific Explanation | Chapter 10 | Research and Applications Towards Mathematics and Computer Science Vol. 2

 This branch presents an iterative solution for the opposite problem of deciding α-order of fractional derivative with regard to occasion for the diffusion equating. The paper discusses the outcomes of the means's numerical use. The computations were done using model issues accompanying precise answers. The study granted the development of correctness studies for questions formulation, examining methods for problems judgment.

Author(s) Details:

A. M. Kardashevsky,
North-Eastern Federal University, Yakutsk, Russia.

Please see the link here: https://stm.bookpi.org/RATMCS-V2/article/view/11113

A New Analysis and Derivations of Properties of Ideal and Fractional Capacitors by Application of New Generalized Formula of Charge Function q(t) = c(t)*v(t)| Chapter 6 | New Frontiers in Physical Science Research Vol. 3

 This episode is continuation of application of a recently developed statement formula for capacitor i.e. charge as a function momentary, which is spiral operation of a time variable capacity function and a period-varying voltage function (various from capacitance multiplied by physical ability to get charge stored in a capacitor). This stage gives a theoretical lawfulness test i.e. analytically acquired in several applications for this new expression. This chapter will suffice in various super-capacitor studies, dielectric entertainment experiments, and impedance spectroscopy for miscellaneous material developments for electrical strength storage responsibilities. This new generalized formula is too verified tentatively. Here we use this new expression and apply to differing types of input excitement voltages those are-sinusoidal, constant step, ramp strength We analyze and define the effects, like the charge, the current, the loss-touching and the memory effects, remembering shape of input potential and extend this to evaluate resistance function of a classical capacitor in addition to a fractional capacitor. We also obtain by using this new rule to get value of equivalent Farads for a partial capacitor having parts in fractional order elaborated on the Nyquist’s drawing.  However, this concept is still to be used to its adequate potential.

Author(s) Details:

Shantanu Das,
Bhabha Atomic Research Center (BARC), Mumbai, India.

Please see the link here: https://stm.bookpi.org/NFPSR-V3/article/view/8592

Review on Fractional Q-calculus with Varying Arguments | Chapter 5 | Recent Advances in Mathematical Research and Computer Science Vol. 5

 We introduce a novel subclass of analytic functions described by fractional q - calculus integral operators in this chapter. Using the concept of q-calculus, we also introduce the q-Bernadi integral operator for analytic functions. For the functions f belonging to the classes V(A, B, q, ) and K(A, B, q, ), we cover coefficient estimates, growth, distortion theorems, and many other features.

Author(S) Details

N. Ravi Kumar
Department of Mathematics, JSS College of Arts, Commerce and Science, Mysore - 570 025, India.

View Book:- https://stm.bookpi.org/RAMRCS-V5/article/view/4967