<p>In this article, we develop a compact finite difference scheme to solve the convection-diffusion-wave equation incorporating the fractional derivative viewed as the Caputo derivative. The numerical discretization of the fractional derivative of order <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$1<\alpha <2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>α</mi> <mo><</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>) is performed with new coefficients <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(i+\frac{1}{2})^{2-\alpha }-(i-\frac{1}{2})^{2-\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mo>-</mo> <mi>α</mi> </mrow> </msup> <mo>-</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mo>-</mo> <mi>α</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> instead of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(i+1)^{2-\alpha }-i^{2-\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mo>-</mo> <mi>α</mi> </mrow> </msup> <mo>-</mo> <msup> <mi>i</mi> <mrow> <mn>2</mn> <mo>-</mo> <mi>α</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> and the space derivatives are substituted with a compact difference scheme of fourth order. The method is proven unconditionally stable. Furthermore, we have shown that the proposed scheme converges with order <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$O(\tau ^{3-\alpha }+h^4)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>τ</mi> <mrow> <mn>3</mn> <mo>-</mo> <mi>α</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>h</mi> <mn>4</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. The scheme’s efficiency is demonstrated through numerical results and plots that authenticate the theoretical results.</p>

A high-order compact numerical algorithm for time-fractional convection-diffusion-wave equation

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发表日期：

2026年3月25日

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Reetika Chawla,
Devendra Kumar

摘要

In this article, we develop a compact finite difference scheme to solve the convection-diffusion-wave equation incorporating the fractional derivative viewed as the Caputo derivative. The numerical discretization of the fractional derivative of order $\alpha $ $α$ ( $1<\alpha <2$ $1 < α < 2$ ) is performed with new coefficients $(i+\frac{1}{2})^{2-\alpha }-(i-\frac{1}{2})^{2-\alpha }$ ${(i + \frac{1}{2})}^{2 - α} - {(i - \frac{1}{2})}^{2 - α}$ instead of $(i+1)^{2-\alpha }-i^{2-\alpha }$ ${(i + 1)}^{2 - α} - i^{2 - α}$ and the space derivatives are substituted with a compact difference scheme of fourth order. The method is proven unconditionally stable. Furthermore, we have shown that the proposed scheme converges with order $O(\tau ^{3-\alpha }+h^4)$ $O (τ^{3 - α} + h^{4})$ . The scheme’s efficiency is demonstrated through numerical results and plots that authenticate the theoretical results.