An integral discretization scheme on a graded mesh for a fractional differential equation with integral boundary conditions
In this paper, a fractional differential equation with integral conditions is studied. The fractional differential equation is transformed into an integral equation with two initial values, where the initial values needs to ensure that the exact solution satisfies the integral boundary conditions. A...
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| Veröffentlicht in: | Journal of mathematical chemistry 2024, Vol.62 (6), p.1384-1398 |
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| container_title | Journal of mathematical chemistry |
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| creator | Cen, Zhongdi Huang, Jian Xu, Aimin |
| description | In this paper, a fractional differential equation with integral conditions is studied. The fractional differential equation is transformed into an integral equation with two initial values, where the initial values needs to ensure that the exact solution satisfies the integral boundary conditions. A graded mesh based on a priori information of the exact solution is constructed and the linear interpolation is used to approximate the functions in the fractional integral. The rigorous analysis about the convergence of the discretization scheme is derived by using the truncation error estimate techniques and the generalized Grönwall inequality. A quasi-Newton method is used to determine the initial values so that the numerical solution satisfies two integral boundary conditions within a prescribed precision. It is shown that the scheme is second-order convergent, which improves the results on the uniform mesh. |
| doi_str_mv | 10.1007/s10910-024-01596-7 |
| format | Article |
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The fractional differential equation is transformed into an integral equation with two initial values, where the initial values needs to ensure that the exact solution satisfies the integral boundary conditions. A graded mesh based on a priori information of the exact solution is constructed and the linear interpolation is used to approximate the functions in the fractional integral. The rigorous analysis about the convergence of the discretization scheme is derived by using the truncation error estimate techniques and the generalized Grönwall inequality. A quasi-Newton method is used to determine the initial values so that the numerical solution satisfies two integral boundary conditions within a prescribed precision. 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The fractional differential equation is transformed into an integral equation with two initial values, where the initial values needs to ensure that the exact solution satisfies the integral boundary conditions. A graded mesh based on a priori information of the exact solution is constructed and the linear interpolation is used to approximate the functions in the fractional integral. The rigorous analysis about the convergence of the discretization scheme is derived by using the truncation error estimate techniques and the generalized Grönwall inequality. A quasi-Newton method is used to determine the initial values so that the numerical solution satisfies two integral boundary conditions within a prescribed precision. It is shown that the scheme is second-order convergent, which improves the results on the uniform mesh.</description><subject>Boundary conditions</subject><subject>Chemistry</subject><subject>Chemistry and Materials Science</subject><subject>Differential equations</subject><subject>Discretization</subject><subject>Exact solutions</subject><subject>Fractional calculus</subject><subject>Integral equations</subject><subject>Math. 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Applications in Chemistry</topic><topic>Mathematical analysis</topic><topic>Original Paper</topic><topic>Physical Chemistry</topic><topic>Quasi Newton methods</topic><topic>Theoretical and Computational Chemistry</topic><topic>Truncation errors</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Cen, Zhongdi</creatorcontrib><creatorcontrib>Huang, Jian</creatorcontrib><creatorcontrib>Xu, Aimin</creatorcontrib><collection>CrossRef</collection><jtitle>Journal of mathematical chemistry</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Cen, Zhongdi</au><au>Huang, Jian</au><au>Xu, Aimin</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>An integral discretization scheme on a graded mesh for a fractional differential equation with integral boundary conditions</atitle><jtitle>Journal of mathematical chemistry</jtitle><stitle>J Math Chem</stitle><date>2024</date><risdate>2024</risdate><volume>62</volume><issue>6</issue><spage>1384</spage><epage>1398</epage><pages>1384-1398</pages><issn>0259-9791</issn><eissn>1572-8897</eissn><abstract>In this paper, a fractional differential equation with integral conditions is studied. The fractional differential equation is transformed into an integral equation with two initial values, where the initial values needs to ensure that the exact solution satisfies the integral boundary conditions. A graded mesh based on a priori information of the exact solution is constructed and the linear interpolation is used to approximate the functions in the fractional integral. The rigorous analysis about the convergence of the discretization scheme is derived by using the truncation error estimate techniques and the generalized Grönwall inequality. A quasi-Newton method is used to determine the initial values so that the numerical solution satisfies two integral boundary conditions within a prescribed precision. It is shown that the scheme is second-order convergent, which improves the results on the uniform mesh.</abstract><cop>Cham</cop><pub>Springer International Publishing</pub><doi>10.1007/s10910-024-01596-7</doi><tpages>15</tpages></addata></record> |
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| subjects | Boundary conditions Chemistry Chemistry and Materials Science Differential equations Discretization Exact solutions Fractional calculus Integral equations Math. Applications in Chemistry Mathematical analysis Original Paper Physical Chemistry Quasi Newton methods Theoretical and Computational Chemistry Truncation errors |
| title | An integral discretization scheme on a graded mesh for a fractional differential equation with integral boundary conditions |
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