BLUES iteration applied to nonlinear ordinary differential equations for wave propagation and heat transfer

Abstract

The iteration sequence based on the BLUES (Beyond Linear Use of Equation Superposition) function method for calculating analytic approximants to solutions of nonlinear ordinary differential equations with sources is elaborated upon. Diverse problems in physics are studied and approximate analytic solutions are found. We first treat a damped driven nonlinear oscillator and show that the method can correctly reproduce oscillatory behaviour. Next, a fractional differential equation describing heat transfer in a semi-infinite rod with Stefan-Boltzmann cooling is handled. In this case, a detailed comparison is made with the Adomian decomposition method, the outcome of which is favourable for the BLUES method. As a final problem, the Fisher equation from population biology is dealt with. For all cases, it is shown that the solutions converge exponentially fast to the numerically exact solution, either globally or, for the Fisher problem, locally.

The BLUES function method for second-order partial differential equations: application to a nonlinear telegrapher equation

“…For nonlinear ODEs with source or sink terms, the BLUES function method has been studied intensively [7,8,9,11] and anticipated in [12]. For PDEs with a first-order time derivative, the role of the source was taken up by the initial condition multiplied by a Dirac point source located at t = 0 [10].…”

“…Recently, it has been demonstrated that the theory of Green functions can be usefully extended to inhomogeneous nonlinear ordinary differential equations (ODEs), effectively using superposition beyond the linear domain [7,8,9]. Consequently, we probed the usefulness of the BLUES function method in the arena of nonlinear partial differential equations (PDEs) with a first-order time derivative where the initial condition plays the role of the inhomogeneous source by multiplication with a Dirac delta point source located at time t = 0 [10].…”

Epidemic processes with vaccination and immunity loss studied with the BLUES function method

“…The zeroth approximant (11) is the convolution of the matrix Green function with the source vector ψ(t), i.e.,…”

“…Here we extend the BLUES iteration originally developed for ordinary DEs [9][10][11] and partial DEs [12] to a system of coupled ordinary DEs. The role of the inhomogeneous source (or sink) term in the context of the ordinary DE will now be taken over by a vector of sources (or sinks).…”

“…There have been attempts to generate approximate solutions for this model using the ADM [6], the VIM and the HPM [7,8] but these solutions diverge quickly and necessitate an improvement of the methods in order to extend the region of convergence. This paper is structured as follows: in Section II we extend the by now established BLUES function method [9][10][11][12] to a system of nonlinear coupled ordinary DEs. Next, in Section III, we describe the SIRS model with a constant vaccination strategy and reduce the system to the two-dimensional subsystem we subsequently solve.…”