In this work, generalize solution based on linear Legendre multi-wavelets are proposed for single, double and triple integrals with variable limits. To obtain the numerical approximations for integrals, an algorithm with the properties of linear Legendre multi-wavelets are applied. The main benefits of this method are its simple applicable and efficient. Furthermore, the error analysis for single, double and triple integrals is worked out to show the efficiency of the method. Numerical examples for the integrals are conducted by using linear Legendre multi-wavelets in order to validate the error estimation.

This research proposes a three-point block method for solving Duffing type higher order ordinary differential equations (ODEs) which is also commonly referred as the Duffing oscillator. The research conducted implements a variable order step size technique for approximating the exact solution for the Duffing Oscillator. The proposed algorithm will be tested against various Duffing oscillators and numerical approximation will be compared with current viable methods. The accuracy and efficiency of the proposed method will be illustrated in the numerical results.