International Journal of Mathematics for Industry

Several well-known nonlinear partial differential equations (NPDEs) of integer order, including the Heat Equation and the Korteweg–de Vries (KDV) Equation, as well as NPDEs of fractional order, including the fractional advection equation (FAE), fractional gas dynamic equation (FGDE), diffusion equation, and fractional Burger equation (FBE), are approximated analytically in this study using the Shehu Transform Adomian Decomposition Method, which is a combination of two methods, namely, Shehu Transform (ST) and Adomian Decomposition Method (ADM). The Caputo sense is utilized to the fractional derivatives. This semianalytical method is used to acquire the series solutions to several example problems. The main advantage of the approach is that, in contrast to other similar semianalytical methods, it is straightforward to apply. This method has been widely used to answer any class of equations in the sciences and engineering because it can solve a huge class of linear and nonlinear equations effectively, more quickly, and precisely. The obtained solution matches exactly with the solution obtained by other methods.

Several significant questions of mathematics and mathematical physics have been effectively explained and answered through the use of fractional kinetic equations containing special functions. Due to the high importance of arbitrary-order kinetic equations, the aim of this work is to obtain the solution of a new arbitrary-order kinetic equation related to the Riemann xi function. The Sumudu transform technique is used to solve it. The findings are plotted with the help of MATLAB R2016a. The outcomes of the paper are in the form of an infinite series representation of the Riemann xi function and in terms of the Mittag-Leffler function.

The Navier–Stokes equation is a key governing equation for the motion of viscous fluid flow. The main target of our work is to obtain the solution to multi-dimensional Navier–Stokes equations in the Yang–Abdel–Cattani fractional sense. The results of the model are in terms of the three-parameter Prabhakar function. Three examples are discussed that depend on the recommended method for a novel Yang–Abdel–Cattani arbitrary order operator. The obtained results are plotted with the help of MATLAB R2016a mathematical software.

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.

In this paper, we have attempted a fresh method to demonstrate how special functions and fractional calculus are used in real-world problems. Here, we have examined the glucose supply in human blood using the incomplete Aleph function (IAF) and the Caputo fractional operator. In this study, we used the incomplete Aleph function to find the blood glucose equation that is supplied to human blood. In terms of various hyper-geometric functions, we have also obtained several significant and unique results, and we have defined the blood glucose function in terms of IAF.

In this paper, we introduce a new approach to the study of finite integral formulas associated with special functions, bringing out the incomplete Aleph functions and other well-known special functions. Our method is based on the finite integral operator applied to incomplete Aleph functions. This approach makes it easier to derive integral formulas that connect the incomplete Aleph functions using well-known special functions and algebraic expressions. As a result, we can group integral formulas and special functions into different classes according to similar features. Furthermore, we can discover new integral formulas with this framework that are useful for computational applications.

This study aims to present the numerical approximations of groundwater flow problem, namely the time-fractional Boussinesq equation using the fractional variational iteration method. Fractional derivatives in time variable are taken in Caputo and Caputo–Fabrizio sense. This work compares the solutions of the time-fractional Boussinesq equation for the Caputo and Caputo–Fabrizio fractional operators. Graphical representation of the water table head’s behavior for different time values is shown.

This paper’s objective is to arrange fractional active conditions while taking into account the Bessel–Maitland capability reported by Khan et al. The solution’s final form is expressed in terms of the Wright hypergeometric function, which allows us to define numerous additional fractional kinetic equation solutions. Generally, fractional kinetic condition is applied to many problems in astronomy and material science.

This study presents a new multivariable fourth class of Horn function. Additionally, for the new extended multivariable fourth-type Horn function, we create bilateral generating functions and particular generating functions for this multivariable fourth-type Horn function. Numerous multilinear and multilateral generating functions are also generated for these functions

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.

Coronavirus spreads worldwide with various symptoms and strains such as COVID-19, SARS, MERS. Outbreak and prevention of the coronavirus can be studied by many mathematical models like SIR, SEIR and SEIQDR. In this paper, we employed Euler’s method based on the SEIQDR mathematical model of COVID-19. Also, after certain iterations of the population categories, we obtained the solution range of the SEIQDR model. In the end, we studied the impact of $\beta$ for integer 1 and fractional order 0.9 and 0.85 on the obtained numerical solutions. The graphical solutions are also analyzed for integer and fractional values of $\beta$. The advantage of this proposed method is that it reduces the mathematical and numerical computations significantly and the proposed method is effective and efficient in extracting the approximate solution of various population categories for COVID-19 mathematical model.