Error estimate and stability analysis on the study of a high-order nonlinear fractional differential equation with Caputo-derivative and integral boundary condition. (English) Zbl 1513.34036
Summary: In this work, we consider a high-order nonlinear fractional differential equation with Caputo-derivative and the boundary condition of integral type which entangles starting point and integration over the domain. It is studied the existence and the uniqueness of the solution. First, the unique exact solution is extracted in terms of Green’s function for the linear fractional differential equation, and then, Banach contraction mapping theorem is applied to prove the main result in the case of general nonlinear source term. The main result is demonstrated by some illustrative examples to show its legitimacy and applicability. To approximate the numerical solution, two algorithms are given. The second one is a kind of PECE (Predict, Evaluate, Correct, Evaluate) method which is based on the piecewise linear interpolation of the source function and applying the Simpson’s quadratic rule to impose nonlocal integral condition. The convergence analysis and the error estimate of the second method are given and its error bound shows that it is of order two. Moreover, the stability analysis reveals that the main algorithm is stable linearly in some sense and the linear stability region is analyzed and discussed. Some numerical experiments are given to illustrate the main numerical procedure and the results are presented graphically.
MSC:
| 34A08 | Fractional ordinary differential equations |
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
| 34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
| 47N20 | Applications of operator theory to differential and integral equations |
| 65L10 | Numerical solution of boundary value problems involving ordinary differential equations |
Keywords:
high-order fractional differential equation; integral boundary condition; Caputo-derivative; fixed point theorem; PECE method; error estimate; linear stabilityReferences:
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