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A space–time spectral Petrov–Galerkin method for nonlinear time fractional Korteweg–de Vries–Burgers equations

In this work, we study a space–time Petrov–Galerkin method for third‐ and fifth‐order time fractional Korteweg–de Vries–Burgers equations. The method is based on the framework of Legendre and Jacobi polynomials. The basis functions of the fractional part are constructed by the generalized Jacobi functions, which contained the singularity of weak solutions. The numerical schemes of the problems are transformed into the nonlinear schemes constructed by matrices. Based on the orthogonality of ideal basis functions, we get the optimal estimation under the specific weighted Sobolev spaces. Numerical experiments confirm the expected convergence.