A hybrid spectral approach based on 2D cardinal and classical second kind Chebyshev polynomials for time fractional 3D Sobolev equation

In this work, the Caputo fractional derivative defines the time fractional 3D Sobolev equation. The 2D shifted second kind Chebyshev cardinal polynomials (SSKCCPs) and 2D shifted second kind Chebyshev polynomials (SSKCPs) (as two well-known classes of basis functions) are utilized to establish a hybrid technique for this new problem. First, the problem solution is approximated simultaneously using the 2D SSKCCPs (relative to x,y$$ x,y $$) and 2D SSKCCPs (relative to z,t$$ z,t $$). Next, the classical and fractional operational matrices of these polynomials are achieved and applied to make the hybrid algorithm. With a combination of derived operational matrices and the collocation approach, solving the expressed fractional 3D problem turns into solving an equivalent system of algebraic equations. The proposed algorithm's accuracy is checked using four numerical examples.

Automated summary

In this study, the Caputo fractional derivative is used to define the time fractional 3D Sobolev equation, and a hybrid technique based on the 2D shifted second kind Chebyshev cardinal polynomials and second kind Chebyshev polynomials is proposed. The problem solution is approximated by simultaneously utilizing these basis functions in two dimensions. Operational matrices of these polynomials are derived and combined with the collocation approach to solve the expressed fractional 3D problem as an equivalent system of algebraic equations. Numerical examples demonstrate the accuracy of the proposed algorithm.