Abstract
A class of non-local boundary value problems for linear fractional differential equations with Caputo-type differential operators is considered. By using integral equation reformulation of the boundary value problem, we study the existence and smoothness of the exact solution. Using the obtained regularity properties and spline collocation techniques, we construct two numerical methods (Method 1 and Method 2) for finding approximate solutions. By choosing suitable graded grids, we derive optimal global convergence estimates and obtain some super-convergence results for Method 2 by requiring additional assumptions on equation and collocation parameters. Some numerical illustrations for verification of theoretical results is also presented.
Highlights
Fractional differential equations arise in various areas of science and engineering and have been proven to be a valuable tool in modelling various phenomena in physics, astrophysics, chemistry, geology, bioengineering, medicine, atmospheric science, material science, optics, mechanics and many other fields
The mathematical aspects of fractional differential equations and various numerical methods for such equations are studied in the monographs [7, 11, 16, 27]
A great deal of papers have been devoted to the numerical solution of initial value problems for fractional differential equations - some more recent results can be found in [6,18,21,23]
Summary
Fractional differential equations arise in various areas of science and engineering and have been proven to be a valuable tool in modelling various phenomena in physics, astrophysics, chemistry, geology, bioengineering, medicine, atmospheric science, material science, optics, mechanics and many other fields. A great deal of papers have been devoted to the numerical solution of initial value problems for fractional differential equations - some more recent results can be found in [6,18,21,23]. We refer to papers [2, 3, 5, 29], which are concerned with various existence results for fractional boundary value problems with non-local conditions. In the present paper we construct two high-order collocation type methods for the numerical solution of the fractional differential equation with non-local boundary values in the following form:. In Theorems 2 and 3 the convergence rates of the proposed algorithms are given
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