Narrow Results

This paper aims to contribute to and broaden the current understanding of fractional calculus and its uses in science. In general, determining the fractional derivative of a product of two functions is an obstacle. In this paper, we overcome this issue for Caputo and Riemann–Liouville fractional derivatives. Specifically, we provide explicit formulas for the derivatives of a product of two functions of order $\alpha \in (0,1)$ for these fractional derivatives. We also provided a few examples to help clarify and bolster our calculations.

The paper presents a computationally efficient method for modeling and simulating distributed systems with lossy transmission line (TL) including multiconductor ones, by a less conventional method. The method is devised based on 1D and 2D Laplace transforms, which facilitates the possibility of incorporating fractional-order elements and frequency-dependent parameters. This process is made possible due to the development of effective numerical inverse Laplace transforms (NILTs) of one and two variables, 1D NILT and 2D NILT. In the paper, it is shown that in high frequency operating systems, the frequency dependencies of the system ought to be included in the model. Additionally, it is shown that incorporating fractional-order elements in the modeling of the distributed parameter systems compensates for losses along the wires, provides higher degrees of flexibility for optimization and produces more accurate and authentic modelling of such systems. The simulations are performed in the Matlab environment and are effectively algorithmized.

The RL and RC circuits are analyzed in this research paper. The classical model of these circuits is generalized using the modern concept of fractional derivative with Mittag-Leffler function in its kernel. The fractional differential equations are solved for exact solutions using the Laplace transform technique and the inverse transformation. The obtained solutions are plotted and presented in tables to show the effect of resistance, inductance and fractional parameter on current and voltage. Furthermore, the statistical analysis is presented to predict the seasonal of time and other parameters on the current flowing in the circuit. The statistical analysis shows that the variation in current is insignificant with respect to time and is more significant with respect to other parameters.

In this paper, we study on the numerical solution of fractional nonlinear system of equations representing the one-dimensional Cauchy problem arising in thermoelasticity. The proposed technique is graceful amalgamations of Laplace transform technique with $q$-homotopy analysis scheme and fractional derivative defined with Atangana–Baleanu (AB) operator. The fixed-point hypothesis is considered in order to demonstrate the existence and uniqueness of the obtained solution for the proposed fractional order model. In order to illustrate and validate the efficiency of the future technique, we consider three different cases and analyzed the projected model in terms of fractional order. Moreover, the physical behavior of the obtained solution has been captured in terms of plots for diverse fractional order, and the numerical simulation is demonstrated to ensure the exactness. The obtained results elucidate that the proposed scheme is easy to implement, highly methodical as well as accurate to analyze the behavior of coupled nonlinear differential equations of arbitrary order arisen in the connected areas of science and engineering.

The main purpose of the present investigation is to find the solution of fractional coupled equations describing the romantic relationships using $q$-homotopy analysis transform method ($q$-HATM). The considered scheme is a unification of $q$-homotopy analysis technique with Laplace transform (LT). More preciously, we scrutinized the behavior of the obtained solution for the considered model with fractional-order, in order to elucidate the effectiveness of the proposed algorithm. Further, for the different fractional-order and parameters offered by the considered method, the physical natures have been apprehended. The obtained consequences evidence that the proposed method is very effective and highly methodical to study and examine the nature and its corresponding consequences of the system of fractional order differential equations describing the real word problems.

The purpose of this research is to analyze the general equations of double diffusive magneto-free convection in a rate type fluid presented in non-dimensional form, and apply to a moving heated vertical plate as in the boundary layer flow up, with existence of externally magnetic fields which are either moving or fixed consistent with the plate. Thermal transport phenomenon is discussed in the presence of constant concentration coupled with first-order chemical reaction with exponential heating. An innovative definition in power law (CF) and Mittag-Leffler (ABC) kernels form time fractional operators are implemented to hypothecate the constitutive mass, heat and momentum equations. The results based on special functions are obtained by using the technique of Laplace transformation to tackle the non-dimensional equations for velocity, mass and energy. The contribution of mass, thermal and mechanical components on the dynamics of fluid is presented and discussed independently. An interesting property regarding the behavior of the fluid velocity is found when the movement is observed in the magnetic intensity along with the plate. In that situation, the fluid velocity is not to be zero when it is far and away from the plate. Moreover, the heat transfer aspects, flow dynamics and their credence on the parameters are drawn out by graphical illustrations along with some special cases for the movement of the plate which are also studied.

The theoretical study focuses on the examination of the convective flow of Oldroyd-B fluid with ramped wall velocity and temperature. The fluid is confined on an extended, unbounded vertical plate saturated within the permeable medium. To depict the fluid flow, the coupled partial differential equations are settled by using the Caputo (C) and Caputo Fabrizio (CF) differential time derivatives. The mathematical analysis of the fractionalized models of fluid flow is performed by Laplace transform (LT). The complexity of temperature and velocity profile is explored by numerical inversion algorithms of Stehfest and Tzou. The fractionalized solutions of the temperature and velocity profile have been traced out under fractional and other different parameters considered. The physical impacts of associated parameters are elucidated with the assistance of the graph using the software MATHCAD 15. We noticed the significant influence of the fractional parameter (memory effects) and other parameters on the dynamics of the fluid flow. Shear stress at the wall and Nusselt number also are considered. It’s brought into notice the fractional-order model (CF) is the best fit in describing the memory effects in comparison to the C model. An analysis of the comparison between the solution of velocity and temperature profile for ramped wall temperature and velocity and constant wall temperature and velocity is also performed.

In this paper, some important objectives have been achieved, which are as follows: First, we present a method of the inverse for a class of non-singular block matrices and some associated properties. Also, the accuracy of a new method is verified with some illustrated examples by applying the MATLAB lines. Second, applying a class of block matrices, we give the exact solution for fractional matrix differential equation systems using the Laplace fractional transformation method. Finally, illustrative examples and individual cases are also presented and discussed to demonstrate our new approach.

Glycolysis, which occurs in the cytoplasm of both prokaryotic and eukaryotic cells, is regarded to be the primary step employed in the breakdown of glucose to extract energy. As it is utilized by all living things on the planet, it was likely one of the first metabolic routes to emerge. It is a cytoplasmic mechanism that converts glucose into carbon molecules while also producing energy. The enzyme hexokinase aids in the phosphorylation of glucose. Hexokinase is inhibited by this mechanism, which generates glucose-6-P from adenosine triphosphate (ATP). This paper’s primary goal is to quantitatively examine the general reaction–diffusion Glycolysis system. Since the Glycolysis model shows a positive result as the unknown variables represent chemical substance concentration, therefore, to evaluate the behavior of the model for the non-integral order derivative of both independent variables, we expanded the concept of the conventional order Glycolysis model to the fractional Glycolysis model. The nonlinearity of the model is decomposed through an Adomian polynomial for evaluation. More precisely, we used the iterative Laplace Adomian decomposition method (LADM) to determine the numerical solution for the underlying model. The model’s necessary analytical/numerical solution was found by adding the first few iterations. Finally, we have presented numerical examples and graphical representations to explain the dynamics of the considered model to ensure the scheme’s validity.

We compute the Laplace transforms of some integral functionals of the three-dimensional Bessel process in terms of modified Bessel functions, Gauss’ hypergeometric functions, and confluent hypergeometric functions. Some new results are obtained, and several established results, such as Dufresne’s perpetuity and a particular case of its translated version, are recovered. In particular, we derive the Laplace transform of the two-dimensional random variable $({\int }_0^{\infty }{(exp({\rho }_t)-\eta )}^{-1}dt,{\int }_0^{\infty }{(exp({\rho }_t)-\eta )}^{-2}dt),$ where $\rho$ is a three-dimensional Bessel process and $\eta \le 1$. A crucial step of the derivation is to construct a martingale, as performed in the Bass solution of the Skorokhod embedding problem, with respect to an enlarged filtration.

Diabetes Mellitus is a metabolic disorder that occurs when the sugar level in blood is not proper. Diabetes is classified into many types and this classification is based upon the etiology of the disease. This disease is classified into three main types: Type 1 Diabetes Mellitus (T1DM), Type 2 Diabetes Mellitus (T2DM) and Gestational Diabetes Mellitus (GDM). In this paper, our aim is to present a model of Diabetes Mellitus Type ${\mathrm{\text{SEII}}}_T$ by considering treatment and genetic factors. We have formulated the model for the epidemic problem. Here, a numerical algorithm based on Fractional Homotopy Analysis Transform Method (FHATM) is constituted to study the fractional form of the model for Diabetes Mellitus Type ${\mathrm{\text{SEII}}}_T$. The suggested technique is obtained by merging the homotopy analysis, Laplace transform and the homotopy polynomials. The convergence and uniqueness of the solution are also discussed and results are also presented.

The local fractional Korteweg–de Vries equations were considered in this paper. Series solutions for the linear and nonlinear case were examined using the local fractional Laplace transform iterative method. This proposed method is the coupling of the local fractional Laplace transform with the new iterative method. The existence and uniqueness of the solutions is considered using the principle of mathematical induction. Furthermore, illustrative examples were considered and the graphs are shown. The results obtained in this study reveal the benefit of this method and provide valuable insight into the behavior of complex phenomena in a precise and efficient manner with less computational work and implementation ease.

This paper expressed the fractional kinetic equation (KE) in terms of the incomplete $\aleph$-function, I-function, Fox H-function, and Meijer’s G-function. It assesses the importance of fractional KE in diverse scientific and engineering contexts, highlighting its relevance across various scenarios. In this paper, we investigate the results of Katugampola’s kinetic fractional equations with the product of the generalized M-series and incomplete $\aleph$-function. The $\tau$-Laplace transform is applied for obtaining the desired results. Utilizing an M-series, the universality of this series is harnessed to deduce solutions for a fractional KE. Furthermore, a graphical representation of the obtained solutions’ behaviors is presented, enhancing the impact of the findings.