Variational Iteration Method For Solving Periodic Boundary Value Problems

Generally speaking, it is difficult for us to solve the periodic boundary value problems of differential equation, especially for high order and nonlinear cases. Since it has been widely used in various fields, finding numerical solutions or approximation solutions becomes more and more important. In this paper, a new kind of analytical technique called the variational iteration method is described and used to give approximation solutions for periodic boundary value problems.The variational iteration method , which has been proved by many authors to be a powerful mathematical tool for various kinds of nonlinear problems, was proposed originally by He [1] in 1999. He surveyed some basic concepts and new nomenclatures, such as restricted variation, correction functional in He[2] in 2005. It was successfully applied to delay differential equations [3], to autonomous ordinary differential systems [4], to Burger's and coupled Burger's equations [9], to Helmholtz equation [12], to nonlinear differential equations of fractional order [14], to two-point boundary value problems [15], and other problems. For linear problems, its exact solution can be obtained by only one iteration due to the fact that Lagrange multiplier can be exactly identified. For nonlinear problems, it is possible to derive the first approximation. Now the variation iteration method will be investigated in details and efficiency of the approach will be shown by applying the procedure on some specified differential equations with periodic boundary value conditions.Consider the forth order linear boundary value problemx⁽⁴⁾ - x" = sint,with boundary conditions x(0) = x(2Ï€),x'(0) = x'(2Ï€),x"(0) = x"(2Ï€), x⁽³⁾(0) = x⁽³⁾(2Ï€). Its correction functional can be written down as follows:whereλ(Ï„, t) is Lagrange multiplier, which can be identified optimally via the variational theory. Taking variation with respect to the independent variable x_n, noticing thatδx_n(0) = 0, for all variationsδx_n,δx'_n,δx"_n andδx_n⁽³⁾ , implying the following stationary conditions:The general Lagrange multiplier,therefore, can be readily identifiedas a result,we obtain the following iteration formula:If we use its complementary solution x₀(t) = ae^t+be^-t+ct+d as an initial approximation, using the iteration formula, we getBy imposing the boundary conditions yields a = -1/4,b=1/4, c = 1 and d∈C, as a result, we have x₁(t) = 1/2 sint + d which is the exact solution.It can be seen clearly that the variation iteration method is powerful and efficient approach from this example.For nonlinear problems, in order to determine the Lagrange multiplier in as simple a manner as possible, the nonlinear terms have to be consider as restricted variations, so the above discussed case also can apply to nonlinear problems, details are demonstrated in the paper. But we need to point out that the lesser the application of restricted variations the faster the approximation converging to its exact solution.We can conclude that a correction functional can be easily construct by generally Lagrange multiplier, and the multiplier can be identified by variation theory. The application of restricted variations in correction functional makes it much easier to determine the multiplier. The initial approximation can be freely selected with unknown constants, which can be determined via various methods. It is convenient for us to iterating many times to arrive the given accuracy for the advanced computer technique. Finally you will find that the method has many merits.