In recent years, the Adomian decomposition method, that can be applied for various types of ordinary and partial differential equations is continuing intensely. The objective of this method is to make possible physically realistic approximate solutions of linear and nonlinear differential equations without discretization. The Adomian decomposition method gives the solution in the form of a infinite series in which the terms are calculated recursively using special polynomials, so-called Adomian polynomials. In this work, we present a modified Adomian decomposition method for solving of two-interval boundary-value problems together with additional transmission conditions at the point of interaction. Consider the following two-interval differential equation
y'' (x)+xy' (x)+y(x)=0, x∈[0,1/2)∪(1/2,1]
subject to the boundary conditions
and additional transmission conditions at the interior singular point x=1/2, given by
y(1/2-0)-βy(1/2+0)=0, y' (1/2-0)-γy' (1/2+0)=0.
Such type of problems arise as a mathematical model of many processes in mechanics, physics and other branches of natural science. For example, they describe the vibrations of a loaded string and the vibrational modes of various systems.
Keywords: Adomian decomposition method, transmission conditions, singular point.
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Anahtar Kelimeler: Adomian decomposition method, Transmission conditions, Singular point