Numerical solution of generalized fractional Volterra integro-differential equations via approximation the Bromwich integral

In this paper, we consider the generalized fractional Volterra integro-differential equations with the regularized Prabhakar derivative and represent the solution of this type of equations in the form of Bromwich integral in the complex plane. Then we select the hyperbolic contour as an optimal contour to approximate the Bromwich integral. Further, an example to show absolute errors for various parameters by using our numerical scheme on hyperbolic contour is given.

Numerical Approximation of Riccati Fractional Differential Equation in the Sense of Caputo-Type Fractional Derivative

The Riccati differential equation is a well-known nonlinear differential equation and has different applications in engineering and science domains, such as robust stabilization, stochastic realization theory, network synthesis, and optimal control, and in financial mathematics. In this study, we aim to approximate the solution of a fractional Riccati equation of order 0<β<1 with Atangana–Baleanu derivative (ABC). Our numerical scheme is based on Laplace transform (LT) and quadrature rule. We apply LT to the given fractional differential equation, which reduces it to an algebraic equation. The reduced equation is solved for the unknown in LT space. The solution of the original problem is retrieved by representing it as a Bromwich integral in the complex plane along a smooth curve. The Bromwich integral is approximated using the trapezoidal rule. Some numerical experiments are performed to validate our numerical scheme.

Exact Solution of Two-Dimensional Unsteady Airflow in an Inclined Trachea

<p>The exact solution of the two-dimensional momentum and continuity equations governing the unsteady airflow in an inclined trachea was carried out. The oscillating airflow was described by setting one side of the boundaries be a periodic pressure function. The solution was obtained using Bromwich integral. The present results show that the inclination angle θ and gravitational force g reduces the pressure drop and has a greater effect on the velocity profile of airflow from the trachea to the lungs. Therefore, sleeping in a horizontal position may lead to negative effects for many patients. However, the inclined position is better, and it is dependent on the slope angle position taking, the benefits of gravitational force and utilizing it to improve the breathing of the body. These results are of help in breathing problems, mild sleep apnea, snoring, asthma and chronic obstructive pulmonary disease.</p>

Dynamic Analysis of Heat Transfer Through a Fin of Constant Cross-Sectional Area: Specified Fin Tip Temperature

A dynamic analysis is performed for heat transfer through a fin with constant cross-sectional area and with a specified fin tip temperature. The process starts with a dynamic energy balance around the fin. Laplace transforms and the Bromwich Integral are used to solve analytically the resulting partial differential equation. The final purely analytical solution is compared to the well known steady state solution. The two match exactly as time approaches infinity. Furthermore it is shown that the steady-state and dynamic characteristics of the fin are directly tied to the Biot number.