The banach contraction principles method for approximate solution of Sine-Gordon equations

This study systematically applies the Banach Contraction Principle Method (BCPM) to obtain highly accurate approximate solutions for the well-known nonlinear Sine-Gordon equation. The proposed analytical method is directly implemented by carefully reformulating the original boundary value problem into an equivalent integral form. Subsequently, we construct successive iterative approximations that mathematically converge to the exact analytical solution under suitable theoretical conditions. To thoroughly validate the accuracy and overall reliability of BCPM, comprehensive numerical examples are presented and examined in detail. The computational results demonstrate remarkably close agreement with exact solutions, particularly for relatively small-time intervals. Ultimately, the method demonstrates superior computational efficiency and numerical stability, even when operating with a significantly limited number of iterations. Consequently, this study establishes BCPM as a highly viable and practical tool for solving complex nonlinear partial differential equations.