In this study, a mathematical model has been developed for the second order hyperbolic one-dimensional time fractional Telegraph equation (TFTE). The fractional derivative has been described in the Caputo sense. The governing equations have been solved by a recent reliable semi-analytic method known as the reduced differential transformation method (RDTM). The method is a powerful mathematical technique for solving wide range of problems. Using RDTM method, it is possible to find exact solution as well as closed approximate solution of any ordinary or partial differential equation. Three numerical examples of TFTE have been provided in order to check the effectiveness, accuracy and convergence of the method. The computed results are also depicted graphically.

A dynamical model of HIV infection of CD4+ T cells is solved numerically using an approximate analytical method so-called the differential transform method (DTM). The solution obtained by the method is an infinite power series for appropriate initial condition, without any discretization, transformation, perturbation, or restrictive conditions. A comparative study between the present method, the classical Euler’s and Runge–Kutta fourth order (RK4) methods is also carried out

In this paper, an implicit logarithmic finite difference method (I-LFDM) is implemented for the numerical solution of one dimensional coupled nonlinear Burgers’ equation. The numerical scheme provides a system of nonlinear difference equations which we linearise using Newton's method. The obtained linear system via Newton's method is solved by Gauss elimination with partial pivoting algorithm. To illustrate the accuracy and reliability of the scheme, three numerical examples are described. The obtained numerical solutions are compared well with the exact solutions and those already available.

In this effort, the consequence of chemical reaction on the thermohaline convection in a permeable layer containing Casson fluid is inspected utilizing both linear and nonlinear stability procedures. To model the momentum equation, the Casson fluid version of Darcy's law is utilized. Applying the linear stability concept, the situation for the start of stationary and oscillatory pattern of convective motion is obtained, whereas the convective heat and mass transferences are determined using nonlinear stability theory. It is found that the oscillatory type of convective flow is probable only if the evaluation of the solutal Rayleigh–Darcy number is less than zero. The effect of the chemical reacting factor, the Casson factor, the Lewis number, and the solutal Rayleigh–Darcy number is to hurry the start of convection toward both stationary and oscillatory styles of convection and also surge the size of the convective cells. The convective heat and mass transfers enhance with increasing the Casson factor, the solutal and thermal Rayleigh–Darcy numbers while both decrease with enhancing the heat capacity quotient. The effect of chemical reacting factor is to increase the mass transmission in the system, while it has no effect on the heat transmission. It is also noted that the convective heat and mass transfers in the system enhance nearly 5.4% and 3.8%, respectively, with a 50% increase in the value of Casson factor.

The impact of chemical reactions on the double-diffusive convective motion in a non-Newtonian viscoelastic fluid (Kuvshiniski type) saturated porous layer is examined applying both linear and nonlinear stability techniques. The Darcy model that includes the Kuvshiniski type viscoelastic effect of viscoelastic fluid and the Boussinesq estimation is employed as the momentum equation. The conditions for the occurrence of the stationary and oscillatory style of convective motions are determined by applying linear stability theory in terms of a critical thermal Rayleigh–Darcy number. Using the weakly nonlinear stability analysis, the convective heat and mass transfers are calculated. It is observed that the occurrence of oscillatory convection is possible only if the value of the solute Rayleigh–Darcy number is negative and also depends on other involved physical parameters. With rising values of the Kuvshiniski parameter and the heat capacity ratio, the range of the solute Rayleigh–Darcy number in which oscillatory convection is privileged diminishes, whereas it grows with the chemical reacting parameter and the Lewis number. The critical thermal Rayleigh–Darcy number at which the convective motion occurs increased nearly 3% with a 15% increase in the value of the Kuvshiniski parameter. Furthermore, the convective heat and mass transfers are reduced by growing the Kuvshiniski parameter and the heat capacity ratio, while both are enhanced by increasing the thermal Rayleigh–Darcy number and the solute Rayleigh–Darcy number.

This article describes a new implicit finite-difference method: an implicit logarithmic finite-difference method (I-LFDM), for the numerical solution of two dimensional time-dependent coupled viscous Burgers’ equation on the uniform grid points. As the Burgers’ equation is nonlinear, the proposed technique leads to a system of nonlinear systems, which is solved by Newton's iterative method at each time step. Computed solutions are compared with the analytical solutions and those already available in the literature and it is clearly shown that the results obtained using the method is precise and reliable for solving Burgers’ equation.