On the Solution of Zabolotskaya–Khokhlov Equation Using Differential Quadrature Method by Fourier and Lagrange Bases

This manuscript provides a comprehensive investigation of the Zabolotskaya–Khokhlov (Z-K) equation, a nonlinear partial differential equation essential for modeling wave propagation in acoustics. The main goal is to formulate and assess a hybrid numerical approach that combines finite difference and differential quadrature techniques, utilizing both Fourier and Lagrange basis functions to improve accuracy and efficiency. The problem formulation commences with defining the parameters and physical meaning of the Z-K equation, succeeded by its discretization in both temporal and spatial domains utilizing the proposed hybrid methodology. We systematically investigate the application of the differential quadrature method for approximating spatial derivatives while utilizing the finite difference method for temporal discretization. The paper introduces a quasi-linearization technique based on Taylor expansion of the nonlinear temporal terms. This dual methodology facilitates a versatile and resilient handling of the nonlinear features of the Z-K equation. A comprehensive error analysis is performed, establishing limits on both temporal and spatial errors related to the numerical solutions. The results indicate that the hybrid approach markedly enhances accuracy relative to conventional methods, with error barriers determined through meticulous mathematical derivation and corroborated by numerical simulations. The proposed scheme provides an error estimate of orders $𝒪\left(\Delta t^2,\frac{\varsigma h^{(N-\sigma )}}{(N-\sigma )!}\right),𝒪\left(\Delta t^2,\frac{\varsigma 2^{(N-\sigma )}}{(N-\sigma )!}\right)$ for both Lagrange and Fourier bases, respectively. The results indicate the method's effectiveness in capturing the complex behavior of wave propagation, rendering it a viable instrument for applications in acoustics and related domains. This study enhances the numerical solutions of the Z-K equation and offers insights into the wider use of hybrid approaches for solving nonlinear wave equations.