On accurate closed-form Coiflet solutions of nonlinear Korteweg–de Vries–Burgers equation for shallow water wave by a time-integration wavelet homotopy method

In the paper, an innovative time-integration homotopy-based wavelet approach has been utilized for obtaining accurate closed-form Coiflet solutions for nonlinear Korteweg–de Vries–Burgers (KdVB) equation, which inherits the merits of Homotopy Analysis Method in dealing with strong nonlinearity and explicit time-integration approach in high precision. The wavelet expansions for spatial dimension at intervals with nonhomogeneous Dirichlet, Neumann, Robin and Cauchy boundaries and time dimension by integration are comprehensively formulated. The explicit iterative wavelet scheme in time integration is proposed by solving linear parabolic or hyperbolic differential equations, while the effectiveness and precision have been verified with good accuracy and efficiency. The nonlinear KdVB equation subjected to various initial and boundary conditions is transformed into a series of degenerated ordinary differential matrix equations further solved by the wavelet scheme, while high-precision closed-form Coiflet solutions for six nonlinear examples are given in excellent agreement with exact ones. Compared with other numerical methods, the time-integration wavelet method is effective for acquisition of accurate Coiflet solutions using fewer computing resource at the same scale of evolution time, which is an alternative and valuable approach for solving nonlinear dispersive–dissipative wave problems.