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, socalled Adomian polynomials. In this work, we present a modified Adomian decomposition method for solving of twointerval boundaryvalue problems together with additional transmission conditions at the point of interaction. Consider the following twointerval differential equation
y'' (x)+xy' (x)+y(x)=0, x∈[0,1/2)∪(1/2,1]
subject to the boundary conditions
y(0)=a, y(1)=b
and additional transmission conditions at the interior singular point x=1/2, given by
y(1/20)βy(1/2+0)=0, y' (1/20)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.
References:
Adomian, G., (1994). Solving Frontier Problems of Physics: The Decomposition Method.
Somali S., Gökmen G., (2007). Adomian Decomposition Method for Nonlinear Sturm
Liouville Problems. Survey in Mathematics and its Applications, ISSN 18426298 Volume 2, 1120.
Yücel, M., Mukhtarov, O. Sh. (2018). A New Treatment of the Decomposition Method for Nonclassical Boundary Value Problems, Journal of Advanced Physics, 7.2, 161166.
Anahtar Kelimeler: Adomian decomposition method, Transmission conditions, Singular point
