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

Table 1

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

NC = 1	𝐿𝑥0^(0)𝐿𝑥=𝑓(𝑡)0(1)=𝑔(𝑡)𝛿(𝑡−𝑡1)𝐿𝑥1^(0)=−𝜔^2𝑥0^(0)2−𝜔^2∫𝑅𝑥0^(1)𝑡1^2𝑑𝑡1𝐿𝑥1^(1)=−2𝜔^2𝑥0^(0)𝑥0^(1)(𝑡1)
NC = 2	𝐿(𝑥2^(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
NC = 3	𝐿𝑥3^(0)=−2𝜔^2𝑥0^(0)𝑥2^(0)−𝜔^2𝑥1^(0)2−2𝜔^2∫𝑅𝑥0^(1)𝑡1𝑥2^(1)𝑡1𝑑𝑡1−𝜔^2∫𝑅𝑥1^(1)𝑡1^2𝑑𝑡1𝐿𝑥3^(1)=−2𝜔^2𝑥0^(0)𝑥2^(1)(𝑡1)−2𝜔^2𝑥1^(0)𝑥1^(1)𝑡1−2𝜔^2𝑥2^(0)𝑥0^(1)𝑡1
NC = 4	𝐿𝑥4^(0)=−2𝜔^2𝑥0^(0)𝑥3^(0)−2𝜔^2𝑥1^(0)𝑥2^(0)−2𝜔^2∫𝑅𝑥0^(1)𝑡1𝑥3^(1)𝑡1𝑑𝑡1−2𝜔^2∫𝑅𝑥1^(1)𝑡1𝑥2^(1)𝑡1𝑑𝑡1𝐿𝑥4^(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
NC = 5	𝐿𝑥5^(0)=−2𝜔^2𝑥0^(0)𝑥4^(0)−2𝜔^2𝑥1^(0)𝑥3^(0)−𝜔^2𝑥2^(0)2−2𝜔^2∫𝑅𝑥0^(1)𝑡1𝑥4^(1)𝑡1𝑑𝑡1−2𝜔^2∫𝑅𝑥1^(1)𝑡1𝑥3^(1)𝑡1𝑑𝑡1−𝜔^2∫𝑅𝑥2^(1)𝑡1^2𝑑𝑡1𝐿𝑥5^(1)=−2𝜔^2𝑥0^(0)𝑥4^(1)𝑡1−2𝜔^2𝑥1^(0)𝑥3^(1)(𝑡1)−2𝜔^2𝑥2^(0)𝑥2^(1)(𝑡1)−2𝜔^2𝑥3^(0)𝑥1^(1)(𝑡1)−2𝜔^2𝑥4^(0)𝑥0^(1)𝑡1