An Efficient Method for Solving Second-Order Fuzzy Order Fuzzy Initial Value Problems

Abstract

In this paper, we present an accurate numerical approach based on the reproducing kernel method (RKM) for solving second-order fuzzy initial value problems (FIVP) with symmetry coefficients such as symmetric triangles and symmetric trapezoids. Finding the exact solution of FIVP is not an easy task since the definition will produce a complicated optimization problem. To overcome this difficulty, a numerical method is developed to solve this type of problems. We start by introducing the necessary definitions and theorems about the fuzzy logic. Then, we derived the kernels for two Hilbert spaces. The RKM is derived for the second-order IVP in the Boolean sense, and then we generalize it for the fuzzy sense. Numerical and theoretical results will be given to obtain the accuracy of the developed technique. We solved four linear and non-linear fuzzy IVPs numerically using the proposed method, and we compute the error in each case to show the efficiency of the method. The absolute error was very small in the four examples.

Keywords

• fuzzy initial value problems
• convergence
• reproducing kernel method