The main objective of this paper is to study the composition continuous Kontorovich-Lebedev wavelet transform (KL-wavelet transform) and wave packet (WPT) based on (KL-transform). Estimates for KL-wavelet are obtained, Plancherel?s relation WPT-transform derived. Reconstruction formula WPT associated KL-transform also deduced at end Calderon?s related using its convolution property obtained.

Two new transforms with piecewise linear kernels are introduced. These analogues of the classical Laplace transform and Z-transform. Properties these investigated applications to ordinary differential equations integral provided. This article is ideal for study as a foundational project in an undergraduate course and/or equations.