Find in Library

We present a finite difference/spectral method for the two-dimensional generalized time fractional cable equation by combining the second-order backward difference method in time and the Galerkin spectral method in space with Legendre polynomials. Through a detailed analysis, we demonstrate that the scheme is unconditionally stable. The scheme is proved to have <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>min</mi><mo>{</mo><mn>2</mn><mo>−</mo><mi>α</mi><mo>,</mo><mspace width="0.166667em"></mspace><mn>2</mn><mo>−</mo><mi>β</mi><mo>}</mo></mrow></semantics></math></inline-formula>-order convergence in time and spectral accuracy in space for smooth solutions, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>,</mo><mspace width="0.166667em"></mspace><mi>β</mi></mrow></semantics></math></inline-formula> are two exponents of fractional derivatives. We report numerical results to confirm our error bounds and demonstrate the effectiveness of the proposed method. This method can be applied to model diffusion and viscoelastic non-Newtonian fluid flow.