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

Table 3

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

NC = 1	𝐿𝑥0^(0)𝐿𝑥=𝑓(𝑡)0(1)=𝑔(𝑡)𝛿(𝑡−𝑡1)2𝐿𝑥0^(2)𝑥=06𝐿0^(3)𝐿𝑥=01(0)=−𝜔^2𝑥0^(0)2−𝜔^2∫𝑅𝑥0^(1)𝑡1^2𝑑𝑡1𝐿𝑥1^(1)=−2𝜔^2𝑥0^(0)𝑥0^(1)𝑡12𝐿𝑥1^(2)=−2𝜔^2𝑥0^(1)𝑡1𝑥0^(1)𝑡26𝐿𝑥1^(3)=0
NC = 2	𝐿𝑥2^(0)=−2𝜔^2𝑥0^(0)𝑥1^(0)−2𝜔^2∫𝑅𝑥0^(1)(𝑡1)𝑥1^(1)(𝑡1)𝑑𝑡1𝐿𝑥2^(1)=−2𝜔^2𝑥0^(0)𝑥1^(1)(𝑡1)−2𝜔^2𝑥1^(0)𝑥0^(1)𝑡1−4𝜔^2∫𝑅𝑥0^(1)(𝑡2)𝑥1^(2)(𝑡1,𝑡2)𝑑𝑡22𝐿𝑥2^(2)=−4𝜔^2𝑥0^(0)𝑥1^(2)(𝑡1,𝑡2)−4𝜔^2𝑥0^(1)(𝑡1)𝑥1^(1)(𝑡2)6𝐿𝑥2^(3)=−12𝜔^2𝑥0^(1)𝑡1𝑥1^(2)𝑡2,𝑡3
NC = 3	𝐿𝑥3^(0)=−2𝜔^2𝑥0^(0)𝑥2^(0)−𝜔^2𝑥1^(0)2−2𝜔^2∫𝑅𝑥0^(1)(𝑡1)𝑥2^(1)(𝑡1)𝑑𝑡1−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)𝑡2𝑥2^(2)𝑡1,𝑡2𝑑𝑡2−4𝜔^2∫𝑅𝑥1^(1)𝑡2𝑥1^(2)𝑡1,𝑡2𝑑𝑡22𝐿𝑥3^(2)=−4𝜔^2𝑥0^(0)𝑥2^(2)𝑡1,𝑡2−4𝜔^2𝑥1^(0)𝑥1^(2)(𝑡1,𝑡2)−4𝜔^2𝑥0^(1)(𝑡1)𝑥2^(1)(𝑡2)−2𝜔^2𝑥1^(1)(𝑡1)𝑥1^(1)(𝑡2)−12𝜔^2∫𝑅𝑥0^(1)(𝑡3)𝑥2^(3)(𝑡1,𝑡2,𝑡3)𝑑𝑡3−8𝜔^2∫𝑅𝑥1^(2)(𝑡1,𝑡3)𝑥1^(2)(𝑡2,𝑡3)𝑑𝑡36𝐿𝑥3^(3)=−12𝜔^2𝑥0^(0)𝑥2^(3)𝑡1,𝑡2,𝑡3−12𝜔^2𝑥0^(1)𝑡1𝑥2^(2)𝑡2,𝑡3−12𝜔^2𝑥1^(1)𝑡1𝑥1^(2)𝑡2,𝑡3
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)(𝑡2)𝑥3^(2)(𝑡1,𝑡2)𝑑𝑡2−4𝜔^2∫𝑅𝑥1^(1)(𝑡2)𝑥2^(2)(𝑡1,𝑡2)𝑑𝑡2−4𝜔^2∫𝑅𝑥2^(1)(𝑡2)𝑥1^(2)(𝑡1,𝑡2)𝑑𝑡2−12𝜔^2∫𝑅𝑥1^(2)(𝑡2,𝑡3)𝑥2^(3)(𝑡1,𝑡2,𝑡3)𝑑𝑡2𝑑𝑡3
	2𝐿𝑥4^(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)𝑡1𝑥2^(1)𝑡2−12𝜔^2∫𝑅𝑥0^(1)(𝑡3)𝑥3^(3)(𝑡1,𝑡2,𝑡3)𝑑𝑡3−12𝜔^2∫𝑅𝑥1^(1)(𝑡3)𝑥2^(3)(𝑡1,𝑡2,𝑡3)𝑑𝑡3−8𝜔^2∫𝑅𝑥1^(2)(𝑡1,𝑡3)𝑥2^(2)(𝑡2,𝑡3)𝑑𝑡3−8𝜔^2∫𝑅𝑥2^(2)(𝑡1,𝑡3)𝑥1^(2)(𝑡2,𝑡3)𝑑𝑡36𝐿𝑥4^(3)=−12𝜔^2𝑥0^(0)𝑥3^(3)(𝑡1,𝑡2,𝑡3)−12𝜔^2𝑥1^(0)𝑥2^(3)(𝑡1,𝑡2,𝑡3)−12𝜔^2𝑥0^(1)(𝑡1)𝑥3^(2)(𝑡2,𝑡3)−12𝜔^2𝑥1^(1)(𝑡1)𝑥2^(2)(𝑡2,𝑡3)−12𝜔^2𝑥2^(1)(𝑡1)𝑥1^(2)(𝑡2,𝑡3)−24𝜔^2∫𝑅𝑥1^(2)(𝑡1,𝑡4)𝑥2^(3)(𝑡2,𝑡3,𝑡4)𝑑𝑡4−24𝜔^2∫𝑅𝑥1^(2)(𝑡2,𝑡4)𝑥2^(3)(𝑡1,𝑡3,𝑡4)𝑑𝑡4−24𝜔^2∫𝑅𝑥1^(2)(𝑡3,𝑡4)𝑥2^(3)(𝑡1,𝑡2,𝑡4)𝑑𝑡4