Fractional Differential Equations

The main objective of this paper is to introduce an analytical study for the water wave solutions of coupled fractional variant Boussinesq equation, which is modelled to investigate the waves in fluid dynamics. Wave transformation in fractional form is applied to convert the original fractional-order nonlinear partial differential equation into another nonlinear ordinary differential equation. The strategy here is to use the unified method to obtain a variety of exact solutions. The unified method works well and reveals distinct exact solutions which are classified into two different types, namely polynomial function and rational function solutions. The results are also depicted graphically for different values of fractional parameter. These findings are highly encouraging and have significant importance for some special physical phenomena in fluid dynamics

New analytical study of water waves described by coupled fractional variant Boussinesq equation in fluid dynamics

In this article, a hybrid technique called homotopy perturbation Elzaki transform method has been applied to solve Navier–Stokes equation of fractional order. In the hybrid technique, homotopy perturbation method and Elzaki transform method are amalgamated. Three example problems are solved with a purpose to validate and demonstrate the efficacy of the present method. It is also demonstrated that the results obtained from the present method are in excellent agreement with the results by other methods. It is shown that the proposed method is found to be reliable, efficient and easy to implement for various related problems of science and engineering.

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Solving time-fractional Navier–Stokes equations using homotopy perturbation Elzaki transform

In this paper we obtain approximate bound state solutions of $N$-dimensional fractional time independent Schrodinger equation for generalised Mie-type potential, namely $V(r^{\alpha})=\frac{A}{r^{2\alpha}}+\frac{B}{r^{\alpha}}+C$. Here $\alpha(0<\alpha<1)$ acts like a fractional parameter for the space variable $r$. When $\alpha=1$ the potential converts into the original form of Mie-type of potential that is generally studied in molecular and chemical physics. The entire study is composed with Jumarie type fractional derivative approach. The solution is expressed via Mittag-Leffler function and fractionally defined confluent hypergeometric function. To ensure the validity of the present work, obtained results are verified with the previous works for different potential parameter configurations, specially for $\alpha=1$. At the end, few numerical calculations for energy eigenvalue and bound states eigenfunctions are furnished for a typical diatomic molecule.

Time independent fractional Schrodinger equation for generalized Mie-type potential in higher dimension framed with Jumarie type fractional derivative

In this paper, we have studied some memory dependent Economic order quantity models as the memory effect has an important role to handle the business policy of the inventory system. Demand of a company product depends on many factors like behavior of staff, environment of the shop, product quality etc which are the main reasons of memory effect on the system. Once the customers gain some poor experience, further they will never purchase products from those companies or shop. So inclusion of memory effect in the inventory model is necessary to handle practical business policy. One of the best way of inclusion of memory effect in the EOQ model is the use of fractional calculus as fractional derivative is defined in terms of integration where the limits of integration are the initial state and current state. Three fractional order models have been developed considering (i) only the rate of change of the inventory level of fractional order  (ii) demand rate as a fractional polynomial of degree 2 ,  where  is the rate of change of the inventory level (iii) demand rate as a fractional polynomial of degree 2m , where m may be different from the order of the rate of change of the inventory level. Here fractional order is physically treated as an index of memory. To develop the models here Caputo type fractional derivative has been applied. Due to solve those problems, we have used primal geometric programming method and finally some numerical examples are cited to establish the memory effect. Our investigation establishes the existence of memory effect on inventory management through fractional formulation of EOQ models which can never be obtained from classical calculus.

/pdf/study-of-memory-effects-in-an-inventory-model-using-1968f3k1qq.pdf

Study of memory effects in an inventory model using fractional calculus

One of the motivations for using fractional calculus in physical systems is due to fact that many times, in the space and time variables we are dealing which exhibit coarse-grained phenomena, meaning that infinitesimal quantities cannot be placed arbitrarily to zero-rather they are non-zero with a minimum length. Especially when we are dealing in microscopic to mesoscopic level of systems. Meaning if we denote x the point in space and t as point in time; then the differentials dx (and dt) cannot be taken to limit zero, rather it has spread. A way to take this into account is to use infinitesimal quantities as (\Deltax)^\alpha (and (\Deltat)^\alpha) with 0 \Deltax. This way defining the differentials-or rather fractional differentials makes us to use fractional derivatives in the study of dynamic systems. In fractional calculus the fractional order trigonometric functions play important role. The Mittag-Leffler function which plays important role in the field of fractional calculus; and the fractional order trigonometric functions are defined using this Mittag-Leffler function. In this paper we established the fractional order Schrodinger equation-composed via Jumarie fractional derivative; and its solution in terms of Mittag-Leffler function with complex arguments and derive some properties of the fractional Schrodinger equation that are studied for the case of particle in one dimensional infinite potential well.

A Study of Fractional Schrodinger Equation-composed via Jumarie fractional derivative

In this paper we have derived the fractional-order Schrodinger equation composed of Jumarie fractional derivative. The solution of this fractional-order Schrodinger equation is obtained in terms of Mittag–Leffler function with complex arguments, and fractional trigonometric functions. A few important properties of the fractional Schrodinger equation are then described for the case of particles in one-dimensional infinite potential well. One of the motivations for using fractional calculus in physical systems is that the space and time variables, which we often deal with, exhibit coarse-grained phenomena. This means infinitesimal quantities cannot be arbitrarily taken to zero – rather they are non-zero with a minimum spread. This type of non-zero spread arises in the microscopic to mesoscopic levels of system dynamics, which means that, if we denote x as the point in space and t as the point in time, then limit of the differentials dx (and dt) cannot be taken as zero. To take the concept of coarse graining into account, use the infinitesimal quantities as (Δx)
 α
 (and (Δt)
 α
 ) with 0 < α < 1; called as ‘fractional differentials’. For arbitrarily small Δx and Δt (tending towards zero), these ‘fractional’ differentials are greater than Δx (and Δt), i.e. (Δx)
 α
 > Δx and (Δt)
 α
 > Δt. This way of defining the fractional differentials helps us to use fractional derivatives in the study of dynamic systems.

A study of fractional Schrödinger equation composed of Jumarie fractional derivative

Klein–Gordon equation is one of the basic steps towards relativistic quantum mechanics. In this paper, we have formulated fractional Klein–Gordon equation via Jumarie fractional derivative and found two types of solutions. Zero-mass solution satisfies photon criteria and non-zero mass satisfies general theory of relativity. Further, we have developed rest mass condition which leads us to the concept of hidden wave. Classical Klein–Gordon equation fails to explain a chargeless system as well as a single-particle system. Using the fractional Klein–Gordon equation, we can overcome the problem. The fractional Klein–Gordon equation also leads to the smoothness parameter which is the measurement of the bumpiness of space. Here, by using this smoothness parameter, we have defined and interpreted the various cases.

Joydip Banerjee

Papers

A Study of Fractional Schrodinger Equation-composed via Jumarie fractional derivative

A study of fractional Schrödinger equation composed of Jumarie fractional derivative

Fractional Klein-Gordon equation composed of Jumarie fractional derivative and its interpretation by a smoothness parameter