A Local Fractional Variational Iteration Method for Laplace Equation within Local Fractional Operators

and Applied Analysis 3 The nonlinear local fractional equation reads as L α u + N α u = 0, (19) where L α and N α are linear and nonlinear local fractional operators, respectively. Local fractional variational iteration algorithm can be written as [37] u n+1 (t) = u n (t) + t0 I t (α) {ξ α [L α u n (s) + N α u n (s)]} . (20) Here, we can construct a correction functional as follows [37]: u n+1 (t) = u n (t) + t0 I t (α) {ξ α [L α u n (s) + N α ?̃? n (s)]} , (21) where ?̃? n is considered as a restricted local fractional variation and ξα is a fractal Lagrange multiplier; that is, δα?̃? n = 0 [37, 40]. Having determined the fractal Lagrangian multipliers, the successive approximations u n+1 , n ≥ 0, of the solution u will be readily obtained upon using any selective fractal function u 0 . Consequently, we have the solution u = lim n→∞ u n . (22) Here, this technology is called the local fractional variational method [37]. We notice that the classical variation is recovered in case of local fractional variation when the fractal dimension is equal to 1. Besides, the convergence of local fractional variational process and its algorithms were taken into account [37]. 4. Solutions to Local Fractional Laplace Equation in Fractal Timespace The local fractional Laplace equation (see [38–40] and the references therein) is one of the important differential equations with local fractional derivatives. In the following, we consider solutions to local fractional Laplace equations in fractal timespace. Case 1. Let us start with local fractional Laplace equation given by ∂ 2α T (x, t) ∂t + ∂ 2α T (x, t) ∂x = 0 (23) and subject to the fractal value conditions