On Solution of Singular Cauchy Problem of Generalized Euler Poisson Darboux Equation Using Extended K-Integral Transmutation Composition Method
OBAT CHARLES ARNEST *
Department of Mathematics, Multimedia University of Kenya, P.O. Box 15653-00503, Nairobi, Kenya.
IYAYA WANJALA
Department of Mathematics, Multimedia University of Kenya, P.O. Box 15653-00503, Nairobi, Kenya.
SITAWA WATTANGA
Department of Mathematics, The Open University of Kenya, P.O. Box 2440 - 00606, Nairobi, Kenya.
*Author to whom correspondence should be addressed.
Abstract
This study solves the initial value problem for the Generalized Euler-Poisson-Darboux (GEPD) equation using analytical approach—the Extended k-Integral Transmutation Composition Method (E-kITCM). This method constructs a new class of transmutation operators based on Extended k-special functions, which are essential for deriving explicit integral solutions to the GEPD equation. The proposed framework expands the applicability of integral transform techniques in handling hyperbolic partial differential equations with singularities. The use of Extended k-functions reflects the equation’s inherent complexity involving generalized special functions. These results have significant implications in wave propagation, quantum mechanics, and the theory of special functions, making them of particular interest to researchers in Mathematics and Physics.
Keywords: Integral transform composition method, singular cauchy generalized euler-poisson-darboux equation, extended k−beta and extended k−gamma function, transmutation operator, extended k-transmutation operator, hankel transform