{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,15]],"date-time":"2026-07-15T13:39:20Z","timestamp":1784122760359,"version":"3.55.0"},"reference-count":0,"publisher":"Walter de Gruyter GmbH","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2015,10,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>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\u2013Liouville 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 <jats:inline-formula id=\"eq1_w2aab2b8b7b1b7b1aab1c13b1b1Aa\">\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/cmam-2015-0022_42256db8a3703121c2b51362f5369c9a.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>O<\/m:mi>\n                              <m:mo>(<\/m:mo>\n                              <m:mi>\u0394<\/m:mi>\n                              <m:msup>\n                                 <m:mi>x<\/m:mi>\n                                 <m:mrow>\n                                    <m:mn>3<\/m:mn>\n                                    <m:mo>-<\/m:mo>\n                                    <m:mi>\u03b1<\/m:mi>\n                                 <\/m:mrow>\n                              <\/m:msup>\n                              <m:mo>)<\/m:mo>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>${O(\\Delta x^{3- \\alpha })}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>,\n<jats:inline-formula id=\"eq2_w2aab2b8b7b1b7b1aab1c13b1b3Aa\">\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/cmam-2015-0022_47b0028685ae548f5a99015349e67658.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mn>1<\/m:mn>\n                              <m:mo>&lt;<\/m:mo>\n                              <m:mi>\u03b1<\/m:mi>\n                              <m:mo>&lt;<\/m:mo>\n                              <m:mn>2<\/m:mn>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>${1&amp;lt;\\alpha &amp;lt;2}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>. 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 <jats:inline-formula id=\"eq3_w2aab2b8b7b1b7b1aab1c13b1b5Aa\">\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/cmam-2015-0022_40c6f59b62421d1ebceeddabafda8aa8.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>O<\/m:mi>\n                              <m:mo>(<\/m:mo>\n                              <m:mi>\u0394<\/m:mi>\n                              <m:mi>t<\/m:mi>\n                              <m:mo>+<\/m:mo>\n                              <m:mi>\u0394<\/m:mi>\n                              <m:msup>\n                                 <m:mi>x<\/m:mi>\n                                 <m:mrow>\n                                    <m:mo movablelimits=\"true\" form=\"prefix\">min<\/m:mo>\n                                    <m:mo>(<\/m:mo>\n                                    <m:mn>3<\/m:mn>\n                                    <m:mo>-<\/m:mo>\n                                    <m:mi>\u03b1<\/m:mi>\n                                    <m:mo>,<\/m:mo>\n                                    <m:mi>\u03b2<\/m:mi>\n                                    <m:mo>)<\/m:mo>\n                                 <\/m:mrow>\n                              <\/m:msup>\n                              <m:mo>)<\/m:mo>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>${O (\\Delta t + \\Delta x^{\\min (3- \\alpha , \\beta )})}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>, <jats:inline-formula id=\"eq4_w2aab2b8b7b1b7b1aab1c13b1b7Aa\">\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/cmam-2015-0022_47b0028685ae548f5a99015349e67658.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mn>1<\/m:mn>\n                              <m:mo>&lt;<\/m:mo>\n                              <m:mi>\u03b1<\/m:mi>\n                              <m:mo>&lt;<\/m:mo>\n                              <m:mn>2<\/m:mn>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>${1&amp;lt; \\alpha &amp;lt;2}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>, <jats:inline-formula id=\"eq5_w2aab2b8b7b1b7b1aab1c13b1b9Aa\">\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/cmam-2015-0022_8bf51a4189219ae82fc897140894a6e7.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>\u03b2<\/m:mi>\n                              <m:mo>&gt;<\/m:mo>\n                              <m:mn>0<\/m:mn>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>${\\beta &amp;gt;0}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>, where <jats:inline-formula id=\"eq6_w2aab2b8b7b1b7b1aab1c13b1c11Aa\">\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/cmam-2015-0022_ed8c4d21708d55a295ea93b70f940358.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>\u0394<\/m:mi>\n                              <m:mi>t<\/m:mi>\n                              <m:mo>,<\/m:mo>\n                              <m:mi>\u0394<\/m:mi>\n                              <m:mi>x<\/m:mi>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>${\\Delta t,\\Delta x}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> 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.<\/jats:p>","DOI":"10.1515\/cmam-2015-0022","type":"journal-article","created":{"date-parts":[[2015,8,20]],"date-time":"2015-08-20T09:03:58Z","timestamp":1440061438000},"page":"497-514","source":"Crossref","is-referenced-by-count":13,"title":["An Algorithm for the Numerical Solution of Two-Sided Space-Fractional Partial Differential Equations"],"prefix":"10.1515","volume":"15","author":[{"given":"Neville J.","family":"Ford","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Chester, CH1 4BJ, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kamal","family":"Pal","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Chester, CH1 4BJ, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yubin","family":"Yan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Chester, CH1 4BJ, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"374","published-online":{"date-parts":[[2015,8,20]]},"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/15\/4\/article-p497.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2015-0022\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2015-0022\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T20:22:13Z","timestamp":1680294133000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2015-0022\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,8,20]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2015,8,28]]},"published-print":{"date-parts":[[2015,10,1]]}},"alternative-id":["10.1515\/cmam-2015-0022"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2015-0022","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,8,20]]}}}