Exponentially fitted non-polynomial cubic spline method for time-fractional singularly perturbed convection-diffusion problems involving large temporal lag

The time-fractional derivative is considered in the Caputo sense and discretized using backward Euler technique. Then, on uniform mesh discretization, a non-polynomial cubic spline scheme is constructed along the spatial direction. To alleviate the effect of the perturbation parameter, an exponential fitting factor is introduced to the scheme. The parameter-uniform convergence of the proposed method is proved rigorously and shown to be <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ε</mml:mi></mml:math> -uniform convergent with order of convergence <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup><mml:mrow><mml:mo>(</mml:mo> <mml:mi>Δ</mml:mi> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo></mml:mrow> <mml:mrow><mml:mn>2</mml:mn> <mml:mo>-</mml:mo> <mml:mi>α</mml:mi></mml:mrow> </mml:msup> <mml:mo>+</mml:mo> <mml:msup><mml:mi>M</mml:mi> <mml:mrow><mml:mo>-</mml:mo> <mml:mn>1</mml:mn></mml:mrow> </mml:msup> <mml:mo>)</mml:mo></mml:mrow> </mml:math> . The validity of the proposed method is tested using model examples and the experimental results are in agreement with the theoretical expectation and produces more accurate solution than some existing methods in the literature.