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The wavelets are important functions in the harmonic analysis. Up to our knowledge, apply wavelets to solve differential equations are limited to ODEs or PDEs with approximate and numerical solutions. In this paper, the novel methods based on the wavelets with two independent variables according to differential invariants are proposed. In fact, the transform groups are constructed that can be acted on the existing solutions and produced the new solution. These groups are based on the symmetry groups (obtained by the Lie symmetry method and other equivalence methods) and mother wavelets. The new method based on the wavelets are presented, new mother wavelets are produced, the corresponding wavelet type transform groups are provided and applied for solving the differential equations. Our method can be used for ODEs and PDEs at every order and give us the analytic solutions according to the existing solutions (from every method without any exception).

Solving Differential Equations by New Wavelet Type Transform Method Based on the Wavelets and Symmetry Groups