| Summary: | In the present work, we investigate the efficiency of a numerical scheme to solve a nonlinear time-fractional heat equation with sufficiently smooth solutions, which was previously reported in the literature [Fract. Calc. Appl. Anal. <b>16</b>: 892–910 (2013)]. In that article, the authors established the stability and consistency of the discrete model using arguments from Fourier analysis. As opposed to that work, in the present work, we use the method of energy inequalities to show that the scheme is stable and converges to the exact solution with order <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo>(</mo><msup><mi>τ</mi><mrow><mn>2</mn><mo>−</mo><mi>α</mi></mrow></msup><mo>+</mo><msup><mi>h</mi><mn>4</mn></msup><mo>)</mo></mrow></semantics></math></inline-formula>, in the case that <inline-formula><math display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula> satisfies <inline-formula><math display="inline"><semantics><mrow><msup><mn>3</mn><mi>α</mi></msup><mo>≥</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></semantics></math></inline-formula>, which means that <inline-formula><math display="inline"><semantics><mrow><mn>0.369</mn><mo>⪅</mo><mi>α</mi><mo>≤</mo><mn>1</mn></mrow></semantics></math></inline-formula>. The novelty of the present work lies in the derivation of suitable energy estimates, and a discrete fractional Grönwall inequality, which is consistent with the discrete approximation of the Caputo fractional derivative of order <inline-formula><math display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula> used for that scheme at <inline-formula><math display="inline"><semantics><msub><mi>t</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msub></semantics></math></inline-formula>.
|
|---|