A power series method for solving ordinary and partial differentials equations motivated by domain growth

TL;DR

The paper presents a unified power-series approach for solving both ODEs and PDEs by decomposing solutions into streams $u^{(n)}$ (or $p_i^{(n)}$) that are generated recursively from initial data. Through examples including a random-walker ODE system, linear wave/diffusion PDEs, and a nonlinear Burgers equation, it demonstrates explicit recurrence relations, closed-form stream expressions, and validation against discrete simulations. It also shows how boundary conditions can be incorporated and discusses extensions to higher dimensions, while acknowledging limitations in convergence analysis and the absence of general closed-form nth-streams for many nonlinear cases. The methodology offers a flexible analytic framework with a tunable parameter $\beta$ to aid numerical stability and positivity, with potential relevance to domain-growth modeling and related transport problems.

Abstract

In this work we present a power series method for solving ordinary and partial differential equations. To demonstrate our method we solve a system of ordinary differential equations describing the movement of a random walker on a one-dimensional lattice, two nonlinear ordinary differential equations, a wave and diffusion equation (linear partial differential equations), and a nonlinear partial differential equation (quasilinear). The inclusion of boundary conditions and the general solutions to other equations of interest are included in the Supplementary material.

Figures (8)

• Figure 1: A one-dimensional lattice with periodic boundary conditions can be represented as a ring. The sites are sequentially labelled from $i \in \{1, 2, ..., N \}$, with $N$ being the total number of sites.
• Figure 2: A comparison of an ensemble average of the discrete model with periodic boundary conditions and Eq. \ref{['eq:metastatic_diffusion_sum']} at different time points. The blue lines indicate the ensemble average from the discrete model and the red lines indicate the solutions of Eq. \ref{['eq:metastatic_diffusion_sum']}. In the discrete model agents were placed from sites 27:33 for the initial condition in each replicate. The ensemble average was calculated from 10000 replicates of the discrete model, and Eq. \ref{['eq:metastatic_diffusion_sum']} was truncated at $n=60$. In the discrete model and Eq. \ref{['eq:metastatic_diffusion_sum']}$P_{m} = 1$ and $\beta = 1$. In (a) $t = 0$, in (b) $t = 50$, and in (c) $t = 200$.
• Figure 3: The streams of sites $i = 25$ and $i=30$ as given by Eq. \ref{['eq:metastatic_diffusion_sol']} for incrementing values of $n$ from 1:60, where $n$ increases from the left to the right in both panels (a) and (b). For both panels (a) and (b) $P_{m} = 1$ and $\beta = 1$. In (a) $i = 25$, and in (b) $i = 30$. The black-dashed line is the sum of all the streams for site $i$, given by Eq. \ref{['eq:metastatic_diffusion_sum']}.
• Figure 4: A comparison of an ensemble average of the discrete model with no-flux boundary conditions and Eqs. \ref{['eq:middle']}, \ref{['eq:bound1gen']} and \ref{['eq:bound2gen']} at different time points. The blue lines indicate the ensemble average and the red lines indicate Eqs. \ref{['eq:middle']}, \ref{['eq:bound1gen']} and \ref{['eq:bound2gen']}. In the discrete model agents were placed from sites 5:20 for the initial condition in each replicate. The ensemble average was calculated from 10000 replicates of the discrete model, and Eqs. \ref{['eq:middle']}, \ref{['eq:bound1gen']} and \ref{['eq:bound2gen']} are truncated at $n = 100$. In the discrete model and Eq. \ref{['eq:metastatic_diffusion_sum']}$P_{m} = 1$. In (a) $t = 0$, in (b) $t = 50$, and in (c) $t = 200$.
• Figure 5: In (a) Eqs. \ref{['eq:nonlin_exp_N0_new1']} and \ref{['eq:nonlin_exp_N0_new4']} are compared with Eq. \ref{['eq:non_lin2_sol']} for $\gamma = 3$ and $C_{1} = 0.1$. The truncation value for Eq. \ref{['eq:nonlin_exp_N0_new4']} in (a) is $n = 10$ and $\beta = 10$. We use symbolic integration in MATLAB to solve Eqs. \ref{['eq:nonlin_exp_N0_new1']} and \ref{['eq:nonlin_exp_N0_new4']}. In (b) Eq. \ref{['eq:non_lin3sol']} is compared with Eq. \ref{['eq:non_lin3anal']} for $y^{0} = 1$ and $\alpha = 1$. The truncation value for Eq. \ref{['eq:non_lin3sol']} in (b) is $n=20$.