Piecewise Approximate Analytical Solutions of High-Order Singular Perturbation Problems with a Discontinuous Source Term

A reliable algorithm is presented to develop piecewise approximate analytical solutions of third- and fourth-order convection diffusion singular perturbation problems with a discontinuous source term. The algorithm is based on an asymptotic expansion approximation and Differential Transform Method (DTM). First, the original problem is transformed into a weakly coupled system of ODEs and a zero-order asymptotic expansion of the solution is constructed. Then a piecewise smooth solution of the terminal value reduced system is obtained by using DTM and imposing the continuity and smoothness conditions. The error estimate of the method is presented. The results show that the method is a reliable and convenient asymptotic semianalytical numerical method for treating high-order singular perturbation problems with a discontinuous source term

Mathematical Modeling of Heat-Transfer for a Moving Sheet in a Moving Fluid

A mathematical model was developed for determining the heat transfer between a moving sheet that passes through a moving fluid environment to simulate the fabrication process of sheet and fiber-like materials. Similarity transformations were introduced to reduce the governing equations to two nonlinear ordinary differential equations. For high values Prandtl number, the energy equation became much stiffer or singularly perturbed and the standard numerical methods failed to handle it. An innovative procedure combining shooting and singular perturbation technique was developed. The results show that the heat transfer depends on the relative velocity between the moving fluid and the moving sheet to a certain value after that value the relative velocity has no effect. If blowing effect is found the thermal layer becomes thinner and temperature profiles are backed together

Re-Evaluating the Classical Falling Body Problem

This paper re-analyzes the falling body problem in three dimensions, taking into account the effect of the Earth’s rotation (ER). Accordingly, the analytic solution of the three-dimensional model is obtained. Since the ER is quite slow, the three coupled differential equations of motion are usually approximated by neglecting all high order terms. Furthermore, the theoretical aspects describing the nature of the falling point in the rotating frame and the original inertial frame are proved. The theoretical and numerical results are illustrated and discussed.The authors would like to thank the referees for their valuable comments and suggestions, which helped to improve the manuscript. Moreover, the first author thanks Prince Sattam bin Abdulaziz University and Deanship of Scientific Research at Prince Sattam bin Abdulaziz University for their continuous support and encouragement.info:eu-repo/semantics/publishedVersio

Piecewise approximate analytical solutions of high-order reaction-diffusion singular perturbation problems with boundary and interior layers

This work aims to present a reliable algorithm that can effectively generate accurate piecewise approximate analytical solutions for third- and fourth-order reaction-diffusion singular perturbation problems. These problems involve a discontinuous source term and exhibit both interior and boundary layers. The original problem was transformed into a system of coupled differential equations that are weakly interconnected. A zero-order asymptotic approximate solution was then provided, with known asymptotic analytical solutions for the boundary and interior layers, while the outer region solution was obtained analytically using an enhanced residual power series approach. This approach combined the standard residual power series method with the Padé approximation to yield a piecewise approximate analytical solution. It satisfies the continuity and smoothness conditions and offers higher accuracy than the standard residual power series method and other numerical methods like finite difference, finite element, hybrid difference scheme, and Schwarz method. The algorithm also provides error estimates, and numerical examples are included to demonstrate the high accuracy, low computational cost, and effectiveness of the method within a new asymptotic semi-analytical numerical framewor

Applications of Adaptive Multi Step Differential Transform Method to Singular Perturbation Problems Arising in Science and Engineering

Abstract: In this paper, piecewise-analytical and numerical solutions of singular perturbation initial-value problems are obtained by an adaptive multi-step differential transform method (MsDTM). The principle of the method is introduced, and then applied to different types of practical problems arising in science and engineering. Analytical and numerical solutions are obtained using piecewise convergent series with easily computable components over a sequence of variable-length sub-intervals. Numerical results are compared to those obtained by the classical MsDTM and the Runge-Kutta method. The results demonstrate the reliability and efficiency of the method in solving the considered problems

Applications of Adaptive Multi Step Differential Transform Method to Singular Perturbation Problems Arising in Science and Engineering

In this paper, piecewise-analytical and numerical solutions of singular perturbation initial-value problems are obtained by an adaptive multi-step differential transform method (MsDTM). The principle of the method is introduced, and then applied to different types of practical problems arising in science and engineering. Analytical and numerical solutions are obtained using piecewise convergent series with easily computable components over a sequence of variable-length sub-intervals. Numerical results are compared to those obtained by the classical MsDTM and the Runge-Kutta method. The results demonstrate the reliability and efficiency of the method in solving the considered problems

Applications of Adaptive Multi Step Differential Transform Method to Singular Perturbation Problems Arising in Science and Engineering

Abstract: In this paper, piecewise-analytical and numerical solutions of singular perturbation initial-value problems are obtained by an adaptive multi-step differential transform method (MsDTM). The principle of the method is introduced, and then applied to different types of practical problems arising in science and engineering. Analytical and numerical solutions are obtained using piecewise convergent series with easily computable components over a sequence of variable-length sub-intervals. Numerical results are compared to those obtained by the classical MsDTM and the Runge-Kutta method. The results demonstrate the reliability and efficiency of the method in solving the considered problems

A New Method for Solving Singularly Perturbed Boundary Value Problems

In this paper, a new initial value method for solving a class of nonlinear singularly perturbed boundary value problems with a boundary layer at one end is proposed. The method is designed for the practicing engineer or applied mathematician who needs a practical tool for these problems (easy to use, modest problem preparation and ready computer implementation). Using singular perturbation analysis the method is distinguished by the following fact: the original problem is replaced by a pair of first order initial value problems; namely, a reduced problem and a boundary layer correction problem. These initial value problems are solved using classical fourth order Runge–Kutta method. Numerical examples are given to illustrate the method. It is observed that the present method approximates the exact solution very well

A New Method for Solving Singularly Perturbed Boundary Value Problems

In this paper, a new initial value method for solving a class of nonlinear singularly perturbed boundary value problems with a boundary layer at one end is proposed. The method is designed for the practicing engineer or applied mathematician who needs a practical tool for these problems (easy to use, modest problem preparation and ready computer implementation). Using singular perturbation analysis the method is distinguished by the following fact: the original problem is replaced by a pair of first order initial value problems; namely, a reduced problem and a boundary layer correction problem. These initial value problems are solved using classical fourth order Runge–Kutta method. Numerical examples are given to illustrate the method. It is observed that the present method approximates the exact solution very well

The Exact Solutions of Fractional Differential Systems with n Sinusoidal Terms under Physical Conditions

This paper considers the classes of the first-order fractional differential systems containing a finite number n of sinusoidal terms. The fractional derivative employs the Riemann–Liouville fractional definition. As a method of solution, the Laplace transform is an efficient tool to solve linear fractional differential equations. However, this method requires to express the initial conditions in certain fractional forms which have no physical meaning currently. This issue formulated a challenge to solve fractional systems under real/physical conditions when applying the Riemann–Liouville fractional definition. The principal incentive of this work is to overcome such difficulties via presenting a simple but effective approach. The proposed approach is successfully applied in this paper to solve linear fractional systems of an oscillatory nature. The exact solutions of the present fractional systems under physical initial conditions are derived in a straightforward manner. In addition, the obtained solutions are given in terms of the entire exponential and periodic functions with arguments of a fractional order. The symmetric/asymmetric behaviors/properties of the obtained solutions are illustrated. Moreover, the exact solutions of the classical/ordinary versions of the undertaken fractional systems are determined smoothly. In addition, the properties and the behaviors of the present solutions are discussed and interpreted