Solving Differential Equations by New Optimized MRA & Invariant Solutions
The wavelets are important functions in the harmonic analysis. Up to our knowledge, apply wavelets to solve differential equations was limited to ODEs or PDEs with approximate and numerical solutions. In this paper, we design father wavelets with two independent variables according to differential invariants and propose the novel method based on the wavelets, make new father wavelets, apply multiresolution analysis (MRA) with these wavelets for solving DEs. Our method can be used for ODEs and PDEs at every order. This method will result in solution in the form of linear combination of father wavelets and corresponding mother wavelets.
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