1,796 research outputs found
Piecewise polynomial collocation for linear boundary value problems of fractional differential equations
AbstractWe consider a class of boundary value problems for linear multi-term fractional differential equations which involve Caputo-type fractional derivatives. Using an integral equation reformulation of the boundary value problem, some regularity properties of the exact solution are derived. Based on these properties, the numerical solution of boundary value problems by piecewise polynomial collocation methods is discussed. In particular, we study the attainable order of convergence of proposed algorithms and show how the convergence rate depends on the choice of the grid and collocation points. Theoretical results are verified by two numerical examples
FdeSolver: A Julia Package for Solving Fractional Differential Equations
Implementing and executing numerical algorithms to solve fractional
differential equations has been less straightforward than using their
integer-order counterparts, posing challenges for practitioners who wish to
incorporate fractional calculus in applied case studies. Hence, we created an
open-source Julia package, FdeSolver, that provides numerical solutions for
fractional-order differential equations based on product-integration rules,
predictor-corrector algorithms, and the Newton-Raphson method. The package
covers solutions for one-dimensional equations with orders of positive real
numbers. For high-dimensional systems, the orders of positive real numbers are
limited to less than (and equal to) one. Incommensurate derivatives are allowed
and defined in the Caputo sense. Here, we summarize the implementation for a
representative class of problems, provide comparisons with available
alternatives in Julia and Matlab, describe our adherence to good practices in
open research software development, and demonstrate the practical performance
of the methods in two applications; we show how to simulate microbial community
dynamics and model the spread of Covid-19 by fitting the order of derivatives
based on epidemiological observations. Overall, these results highlight the
efficiency, reliability, and practicality of the FdeSolver Julia package
High order algorithms for numerical solution of fractional differential equations
This document is the Accepted Manuscript version of a published work that appeared in final form in [Advances in Difference Equations]. To access the final edited and published work see http://dx.doi.org/10.1186/s13662-021-03273-4.In this paper, two novel high order numerical algorithms are proposed for solving fractional differential equations where the fractional derivative is considered in the Caputo sense. The total domain is discretized into a set of small subdomains and then
the unknown functions are approximated using the piecewise Lagrange interpolation polynomial of degree three and degree four. The detailed error analysis is presented, and it is analytically proven that the proposed algorithms are of orders 4 and 5. The stability of the algorithms is rigorously established and the stability region is also achieved. Numerical examples are provided to check the theoretical results and illustrate the efficiency and applicability of the novel algorithms
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