Modified Homotopy Perturbation Method and Approximate Solutions to a Class of Local Fractional Integrodifferential Equations

In this paper, the local fractional version of homotopy perturbation method (HPM) is established for a new class of local fractional integral-differential equation (IDE). With the embedded homotopy parameter monotonously changing from 0 to 1, the special easy-to-solve fractional problem continuously deforms to the class of local fractional IDE. As a concrete example, an explicit and exact Mittag–Leffler function solution of one special case of the local fractional IDE is obtained. In the process of solving, two initial solutions are selected for the iterative operation of local fractional HPM. One of the initial solutions has a critical condition of convergence and divergence related to the fractional order, and the other converges directly to the real solution. This paper reveals that whether the sequence of approximate solutions generated by the iteration of local fractional HPM can approach the real solution depends on the selection of the initial approximate solutions and sometimes also depends on the fractional order of the selected initial approximate solutions or the considered equations.