Solving Differential Equations by New Optimized MRA & Invariant Solutions - Yazdani, Hamid Reza; Nadjakhah, Mehdi (Communications in Mathematics and Applications, 2017)
The wavelets are important functions in the harmonic analysis"Ž. "ŽUp to our knowledge"Ž, "Žapply wavelets to solve differential equations was limited to ODEs or PDEs with approximate and numerical solutions"Ž. "ŽIn this paper"Ž, "Žwe design father wavelets with two independent variables according to differential invariants and propose the novel method based on the wavelets"Ž, "Žmake new father wavelets"Ž, "Žapply multiresolution analysis (MRA) with these wavelets for solving DEs"Ž. "ŽOur method can be used for ODEs and PDEs at every order"Ž. "ŽThis method will result in solution in the form of linear combination of father wavelets and corresponding mother wavelets.