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This aritcle describes a simple alternative unified method of solving nonlinear regular perturbation problems. The procedure is based upon the manipulation of Taylor's approximation for the expansion of the nonlinear term in the perturbed equation. An essential feature of this technique is the relative simplicity used and the associated unified computational procedure that is employed. As such it should be of interest to teachers of applied mathematics courses, particularly those courses which include perturbation methods. One of the merits of this approach is that it leads on naturally to a scheme based on Taylor's expansions and, consequently, allows the regular perturbation method to be introduced into a course much earlier than is currently common. The method is illustrated by implementing it to four perturbation problems, including two algebraic equations and two initial-value problems. The new approach is compared and contrasted with the traditional perturbation scheme in order to demonstrate its relative merit. [ABSTRACT FROM AUTHOR]

In this paper we introduce and describe a new scheme for the numerical integration of smooth functions. The scheme is based on the modified Taylor expansion and is suitable for functions that exhibit near-sinusoidal or repetitive behaviour. We discuss the method and its rate of convergence, then implement it for the approximation of certain integrals. Examples include integrands involving Airy wave, Bessel, Gamma, and elliptic functions. The results, from the data in the tables, demonstrate that the method converges rapidly and approximates the integral as well as some well-known numerical integration methods used with sufficiently small step sizes. [ABSTRACT FROM AUTHOR]

A numerical method based on Laplace transform and a modified decomposition technique is introduced for the approximate solution of Troesch's problem. The underlying idea of the scheme is to assume an infinite series solution and operate with Laplace transform integral operator to both sides of the differential equation. The nonlinear term is then decomposed, and an iterative algorithm is constructed for the determination of the terms of the infinite series solution. Two special cases of the problem are illustrated and the results show that the method is very accurate and converges rapidly. [ABSTRACT FROM AUTHOR]