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• W2174166392 abstract "Abstract We introduce an algorithm for solving two-sided space-fractional partial differential equations. The space-fractional derivatives we consider here are left-handed and right-handed Riemann–Liouville fractional derivatives which are expressed by using Hadamard finite-part integrals. We approximate the Hadamard finite-part integrals by using piecewise quadratic interpolation polynomials and obtain a numerical approximation of the space-fractional derivative with convergence order <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>O</m:mi> <m:mo>(</m:mo> <m:mi>Δ</m:mi> <m:msup> <m:mi>x</m:mi> <m:mrow> <m:mn>3</m:mn> <m:mo>-</m:mo> <m:mi>α</m:mi> </m:mrow> </m:msup> <m:mo>)</m:mo> </m:mrow> </m:math> ${O(Delta x^{3- alpha })}$ , <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mn>1</m:mn> <m:mo><</m:mo> <m:mi>α</m:mi> <m:mo><</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> ${1&lt;alpha &lt;2}$ . A shifted implicit finite difference method is applied for solving the two-sided space-fractional partial differential equation and we prove that the order of convergence of the finite difference method is <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>O</m:mi> <m:mo>(</m:mo> <m:mi>Δ</m:mi> <m:mi>t</m:mi> <m:mo>+</m:mo> <m:mi>Δ</m:mi> <m:msup> <m:mi>x</m:mi> <m:mrow> <m:mo movablelimits=true form=prefix>min</m:mo> <m:mo>(</m:mo> <m:mn>3</m:mn> <m:mo>-</m:mo> <m:mi>α</m:mi> <m:mo>,</m:mo> <m:mi>β</m:mi> <m:mo>)</m:mo> </m:mrow> </m:msup> <m:mo>)</m:mo> </m:mrow> </m:math> ${O (Delta t + Delta x^{min (3- alpha , beta )})}$ , <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mn>1</m:mn> <m:mo><</m:mo> <m:mi>α</m:mi> <m:mo><</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> ${1&lt; alpha &lt;2}$ , <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>β</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> ${beta &gt;0}$ , where <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>Δ</m:mi> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mi>Δ</m:mi> <m:mi>x</m:mi> </m:mrow> </m:math> ${Delta t,Delta x}$ denote the time and space stepsizes, respectively. Numerical examples where the solutions have varying degrees of smoothness are presented and compared with the exact analytical solution to compare the practical performance of the method with the theoretical order of convergence." @default.
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• W2174166392 date "2015-08-20" @default.
• W2174166392 modified "2023-09-25" @default.
• W2174166392 title "An Algorithm for the Numerical Solution of Two-Sided Space-Fractional Partial Differential Equations" @default.
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• W2174166392 doi "https://doi.org/10.1515/cmam-2015-0022" @default.
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