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 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.

阐述了基于BLUES（超越方程组的线性使用）函数方法的迭代序列，用于计算带源的非线性常微分方程解的解析近似值。研究了物理学中的各种问题，并找到了近似的解析解。我们首先处理阻尼驱动的非线性振荡器，并证明该方法可以正确地再现振荡行为。接下来，将处理一个分数阶微分方程，该方程描述了利用Stefan-Boltzmann冷却的半无限杆中的热传递。在这种情况下，使用Adomian分解方法进行了详细的比较，其结果对BLUES方法是有利的。作为最后一个问题，处理了种群生物学的费舍尔方程。对于所有情况，