Numerical solutions for the Sine-Gordon equation based on the Quintic B-Spline approach

This paper proposes a numerical method for solving the sine-Gordon equation using a technique based on quintic B-spline functions. The use of quintic B-splines provides smooth and flexible approximations, making it suitable for accurately representing complex solution behaviors. The method is designed to efficiently capture the key dynamics of the sine-Gordon equation while remaining computationally effective. Stability of the approach is investigated through von Neumann, confirming that the scheme is conditionally stable under certain parameter settings. To validate its performance, two approximate cases are presented, with error norms calculated to assess the precision of the solutions. The numerical outcomes are then compared with analytical solutions and with other existing computational methods, demonstrating the method’s superior accuracy and convergence characteristics. Overall, the proposed quintic B-spline scheme provides a reliable and efficient approach to numerical simulation of nonlinear wave phenomena, offering valuable potential for applications in applied mathematics, physics, and engineering problems involving nonlinear partial differential equations.