Numerical Approximation of Higher-Order Solutions of the Quadratic Nonlinear Stochastic Oscillatory Equation Using WHEP Technique

Table 2

The equivalent deterministic systems for second-order approximation with different number of corrections (NC).

NC = 1	𝐿𝑥0^(0)=𝑓(𝑡)
	𝐿𝑥0^(1)=𝑔(𝑡)𝛿𝑡−𝑡1
	2𝐿𝑥0^(2)=0
	𝐿𝑥1^(0)=−𝜔^2𝑥0^(0)2−𝜔^2∫𝑅𝑥0^(1)𝑡1^2𝑑𝑡1
	𝐿𝑥1^(1)=−2𝜔^2𝑥0^(0)𝑥0^(1)𝑡1
	2𝐿𝑥1^(2)=−2𝜔^2𝑥0^(1)𝑡1𝑥0^(1)𝑡2
NC = 2	𝐿𝑥2^(0)=−2𝜔^2𝑥0^(0)𝑥1^(0)−2𝜔^2∫𝑅𝑥0^(1)𝑡1𝑥1^(1)𝑡1𝑑𝑡1
	𝐿𝑥2^(1)=−2𝜔^2𝑥0^(0)𝑥1^(1)𝑡1−2𝜔^2𝑥1^(0)𝑥0^(1)𝑡1−4𝜔^2∫𝑅𝑥0^(1)𝑡2𝑥1^(2)𝑡1,𝑡2𝑑𝑡2
	2𝐿𝑥2^(2)=−4𝜔^2𝑥0^(0)𝑥1^(2)𝑡1,𝑡2−4𝜔^2𝑥0^(1)𝑡1𝑥1^(1)𝑡2
NC = 3	𝐿𝑥3^(0)=−2𝜔^2𝑥0^(0)𝑥2^(0)−𝜔^2𝑥1^(0)2−2𝜔^2∫𝑅𝑥0^(1)𝑡1𝑥2^(1)𝑡1𝑑𝑡1−𝜔^2∫𝑅𝑥1^(1)𝑡1^2𝑑𝑡1−2𝜔^2∫𝑅2𝑥1^(2)𝑡1,𝑡2^2𝑑𝑡1𝑑𝑡2𝐿𝑥3^(1)=−2𝜔^2𝑥0^(0)𝑥2^(1)(𝑡1)−2𝜔^2𝑥1^(0)𝑥1^(1)(𝑡1)−2𝜔^2𝑥2^(0)𝑥0^(1)(𝑡1)−4𝜔^2∫𝑅𝑥0^(1)(𝑡2)𝑥2^(2)(𝑡1,𝑡2)𝑑𝑡2−4𝜔^2∫𝑅𝑥1^(1)(𝑡2)𝑥1^(2)(𝑡1,𝑡2)𝑑𝑡22𝐿𝑥3^(2)=−4𝜔^2𝑥0^(0)𝑥2^(2)(𝑡1,𝑡2)−4𝜔^2𝑥1^(0)𝑥1^(2)(𝑡1,𝑡2)−4𝜔^2𝑥0^(1)𝑡1𝑥2^(1)𝑡2−2𝜔^2𝑥1^(1)𝑡1𝑥1^(1)𝑡2−8𝜔^2∫𝑅𝑥1^(2)𝑡1,𝑡3𝑥1^(2)𝑡2,𝑡3𝑑𝑡3
NC = 4	𝐿𝑥4^(0)=−2𝜔^2𝑥0^(0)𝑥3^(0)−2𝜔^2𝑥1^(0)𝑥2^(0)−2𝜔^2∫𝑅𝑥0^(1)(𝑡1)𝑥3^(1)(𝑡1)𝑑𝑡1−2𝜔^2∫𝑅𝑥1^(1)(𝑡1)𝑥2^(1)(𝑡1)𝑑𝑡1−4𝜔^2∫𝑅2𝑥1^(2)(𝑡1,𝑡2)𝑥2^(2)(𝑡1,𝑡2)𝑑𝑡1𝑑𝑡2𝐿𝑥4^(1)=−2𝜔^2𝑥0^(0)𝑥3^(1)𝑡1−2𝜔^2𝑥1^(0)𝑥2^(1)𝑡1−2𝜔^2𝑥2^(0)𝑥1^(1)𝑡1−2𝜔^2𝑥3^(0)𝑥0^(1)𝑡1−4𝜔^2∫𝑅𝑥0^(1)𝑡2𝑥3^(2)𝑡1,𝑡2𝑑𝑡2−4𝜔^2∫𝑅𝑥1^(1)𝑡2𝑥2^(2)𝑡1,𝑡2𝑑𝑡2−4𝜔^2∫𝑅𝑥2^(1)𝑡2𝑥1^(2)𝑡1,𝑡2𝑑𝑡22𝐿𝑥4^(2)=−4𝜔^2𝑥0^(0)𝑥3^(2)𝑡1,𝑡2−4𝜔^2𝑥1^(0)𝑥2^(2)(𝑡1,𝑡2)−4𝜔^2𝑥2^(0)𝑥1^(2)(𝑡1,𝑡2)−4𝜔^2𝑥0^(1)(𝑡1)𝑥3^(1)(𝑡2)−4𝜔^2𝑥1^(1)(𝑡1)𝑥2^(1)(𝑡2)−8𝜔^2∫𝑅𝑥1^(2)(𝑡1,𝑡3)𝑥2^(2)(𝑡2,𝑡3)𝑑𝑡3−8𝜔^2∫𝑅𝑥2^(2)(𝑡1,𝑡3)𝑥1^(2)(𝑡2,𝑡3)𝑑𝑡3