A new algorithm for fractional differential equation based on fractional order reproducing kernel space

• Published in: Mathematical methods in the applied sciences 2021-01, Vol.44 (2), p.2171-2182
• Main Authors: Zhang, Ruimin, Lin, Yingzhen
• Format: Article
• Language: English
• Subjects: Algorithms; Complexity; complexity analysis; Convergence; convergence order; Differential equations; Exact solutions; fractional reproducing kernel space; initial value problem of fractional equations; Kernels; Mathematical analysis; Polynomials
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.6927

This paper develops an effective and new method to solve a class of fractional differential equations. The method is based on a fractional order reproducing kernel space. First, depending on some theories, a fractional order reproducing kernel space Wα[0, 1] is constructed. The fractional order reproducing kernel space is a very suitable space to solve a class of fractional differential equations. Then, we calculate the reproducing kernel Ry(x) of the space Wα[0, 1] skilfully in §3. And convergence order and time complexity of this algorithm are discussed. We prove that the approximate solution un of (1.1) converges to its exact solution u is not less than the second order. The time complexity of the algorithm is equal to the polynomial time of the third degree. Finally, three experiments support the algorithm strongly from the aspect of theory and technique.