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A new integral transform on time scales and its applications. (English) Zbl 1294.34083

Summary: Integral transform methods are widely used to solve the several dynamic equations with initial values or boundary conditions which are represented by integral equations. With this purpose, the Sumudu transform is introduced in this article as a new integral transform on a time scale \(\mathbb T\) to solve a system of dynamic equations. The Sumudu transform on time scale \(\mathbb T\) has not been presented before. The results in this article not only can be applied to ordinary differential equations when \(\mathbb T=\mathbb R\), difference equations when \(\mathbb T=\mathbb N_0\), but also, can be applied to \(q\)-difference equations when \(\mathbb T=q^{\mathbb N_0}\), where \(q^{\mathbb N_0}:=\{q^t: t\in\mathbb N_0\) for \(q>1\}\) or \(\mathbb T= q^{\overline{\mathbb Z}}:= q^{\mathbb Z}\cup\{0\}\) for \(q>1\) (which has important applications in quantum theory) and on different types of time scales like \(\mathbb T=h\mathbb N_0,\;\mathbb T=\mathbb N^2_0\) and \(\mathbb T=\mathbb T_n\) the space of the harmonic numbers. Finally, we give some applications to illustrate our main results.

MSC:

34N05 Dynamic equations on time scales or measure chains
44A15 Special integral transforms (Legendre, Hilbert, etc.)
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