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 behavior. Next, a fractional differential equation describing heat transfer in a semi-infinite rod with StefanβBoltzmann 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.
Highlights
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
While linear Differential equations (DEs) with sources can be treated by the theory of Green functions and the principle of superposition, nonlinear DEs violate the superposition principle and a solution cannot be constructed in this way
We present a detailed comparison with an alternative method, the Adomian decomposition method (ADM)
Summary
The usefulness was demonstrated of simple exponential tail solutions of nonlinear reactiondiffusion-convection DEs describing traveling wave fronts. These simple solutions are exact pte provided a co-moving Dirac delta function source is added to the DE. In [9] it was observed that for some problems one and the same simple exponential tail solution simultaneously solves the nonlinear DE with a Dirac delta source as well as a related linear DE with the same Dirac delta source This observation led to the idea to formulate an analytic method ce that uses the concept of Green function, and its convolution with an arbitrary source, beyond the linear domain.
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