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JOURNALS // Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences // Archive

Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2014 Issue 1(34), Pages 19–24 (Mi vsgtu1265)

This article is cited in 5 papers

Differential Equations

Generalized Integral Laplace Transform and Its Application to Solving Some Integral Equations

S. M. Zaikinaab

a Volgograd State University, Volgograd, 400062, Russian Federation
b Samara State Technical University, Samara, 443100, Russian Federation

Abstract: We present integral transforms $\widetilde {\mathcal L}\left\{f(t);x\right\}$ and $\widetilde {\mathcal L}_{\gamma_1,\gamma_2,\gamma} \left\{f(t);x\right\}$, generalizing the classical Laplace transform. The $(\tau, \beta)$- generalized confluent hypergeometric functions are the kernels of these integral transforms. At certain values of the parameters these transforms coincides with the famous classical Laplace transform. The inverse formula for the transforms is given. The convolution theorem for transform $\widetilde {\mathcal L}\left\{f(t);x\right\}$ is proven. Volterra integral equations of the first kind with core containing the generalized confluent hypergeometric function ${\mathstrut}_1\Phi{\mathstrut}_1^{\tau,\beta}(a;c;z)$ are considered. The above equation is solved by the method of integral transforms. The treatment of integral transforms is applied to get the desired solution of the integral equation. The solution is obtained in explicit form.

Keywords: Laplace integral transform, integral equations, generalized hypergeometric function.

UDC: 517.442+517.581

PACS: 075

MSC: Primary 44A10, 44A20; Secondary 33C15, 33C20

Original article submitted 30/IX/2013
revision submitted – 05/XII/2013

DOI: 10.14498/vsgtu1265



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