Two linearized difference schemes on graded meshes for the time-space fractional nonlinear diffusion-wave equation with an initial singularity

Abstract In this article, two linearized difference schemes are considered to solve a class of time-space fractional nonlinear diffusion-wave equations with weak singularity near the initial time based on its equivalent integro-differential equation. The Crank-Nicolson method combined with the trapezoidal product integration rule with graded meshes is implemented to approximate the Riemann-Liouville integral. The fractional central difference formula and a novel compact difference method are used to construct fully discrete difference schemes. Using the energy method, the stability and convergence of the proposed schemes are demonstrated in terms of the L 2 norm with the accuracy of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic">O</mml:mi> <mml:mfenced close=")" open="("> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>N</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mi>min</mml:mi> <mml:mo stretchy="false">{</mml:mo> <mml:mi>r</mml:mi> <mml:mi>σ</mml:mi> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">}</mml:mo> </mml:mrow> </mml:msup> <mml:mo>+</mml:mo> <mml:msup> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:mfenced> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic">O</mml:mi> <mml:mfenced close=")" open="("> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>N</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mi>min</mml:mi> <mml:mo stretchy="false">{</mml:mo> <mml:mi>r</mml:mi> <mml:mi>σ</mml:mi> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">}</mml:mo> </mml:mrow> </mml:msup> <mml:mo>+</mml:mo> <mml:msup> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:mfenced> </mml:math> , respectively. The theoretical analysis is complemented by numerical results.