Approximation of a function with bounded derivatives of first and second order by the extended Sine-Cosine wavelet expansion with applications

Wavelets are very powerful tools for solving certain problems in mathematical analysis. Dueto their well localized behavior, wavelets are very useful for developing new numerical methodsand due to this reason researchers are trying to develop new numerical techniques using differentwavelets. Keeping it mind, In this paper, we have introduced the extended sine-cosine waveletand it is used to find the approximations of a functions having bounded derivatives upto thesecond order. Next, we have calculated the operational matrix of integration for different valuesof parameter µ using these approximations. Then, we have applied these approximations andoperational matrices to find the solutions of some differential and integral equations. Lastly, thecomparison between exact solution and approximate solutions have been discussed to show theusefulness of the method. From the tables 1 and 3, we see that as we increase the value of µ,the approximate solution becomes closer to the exact solution which shows the validity of theproposed method