Generalized Hermite Spectral Method Matching Different Algebraic Decay at infinities

In this paper, we propose a new generalized Hermite spectral method. We introduce an orthogonal family of new generalized Hermite functions, with the weight function $$(1+\frac{2}{\pi } \arctan x)^{\alpha }(1-\frac{2}{\pi }\arctan x)^{\gamma }$$(1+2πarctanx)α(1-2πarctanx)γ, $$\alpha $$α and $$\gamma $$γ being arbitrary real numbers. The basic results on the corresponding orthogonal approximation and interpolation are established. As examples of applications, we provide the spectral schemes for a linear problem and the Fisher equation, which possess the spectral accuracy in space and match the different algebraic decay at infinities reasonably. Numerical results demonstrate their high efficiency and coincide well with the analysis.