A new shifted generalized Chebyshev approach for multi-dimensional sinh-Gordon equation

Abstract The numerical treatment of multi-dimensional non-linear sinh-Gordon equations is the focus of this paper. We numerically solve the (1 + 1) and (2 + 1) sinh-Gordon equations using two collocation algorithms. We select the set of basis functions as a set of generalized Chebyshev polynomials (CPs), which we express as orthogonal combinations of CPs. We develop and utilize some formulas related to these polynomials to propose our numerical algorithms. Specific values for the high-order derivatives of the basis functions serve in the derivation of the two presented algorithms. Additionally, we provide an estimation of the basis functions used in the convergence analysis study. We follow the two collocation algorithms to transform the sinh-Gordon equations into non-linear equation systems, which any suitable solver can handle. We provide some examples and comparisons to confirm the effectiveness of our presented algorithms.