Reproducing kernel method for the numerical solution of the sine-Gordon equation with initial conditions

In this work, we consider two numerical solution approaches for the sine-Gordon equation with initial conditions using the reproducing kernel Hilbert space method. The solution is represented in the form of a series contained in the reproducing kernel space. In the first approach, we avoid having to perform the Gram-Schmidt orthogonalization process on the basis functions, and this will decrease the computational time. Meanwhile, in the second approach, working with orthonormal basis elements gives some numerical advantages, despite the increased computational time. Applicability and accuracy of the method is examined on an example.