Partial differential equations with matricial coefficients and generalized translation operators

Abstract

Abstract. Let ∆α be the Bessel operator with matricial coefficients defined on (0, ∞) bywhere α is a diagonal matrix and let q be an n × n matrix-valued function.In this work, we prove that there exists an isomorphism X on the space of even C ∞ , C n -valued functions which transmutes ∆α and (∆α + q). This allows us to define generalized translation operators and to develop harmonic analysis associated with (∆α + q). By use of the Riemann method, we provide an integral representation and we deduce more precise information on these operators.

Transmutation Operator Method for Solving Heat Conduction Problem

“…Comparing formulas (15) and (20), we conclude the advantage of transmutation operator's method. In formula (15), the solution is given by a double integral, and in formula (20) it is need to use four -times integral.…”

“…Comparing formulas (15) and (20), we conclude the advantage of transmutation operator's method. In formula (15), the solution is given by a double integral, and in formula (20) it is need to use four -times integral. Thus, the transmutation operator's method has all the advantages of the integral transforms method and is more efficient from a computational point of view.…”

“…In works [3,4,5,7,14,16,18,20] the theoretical foundations of transmutation operator's method are created. Papers [8,[10][11][12][13]15,17,19] are devoted to various applications of the method, including the solving of partial differential equations. The most important of the applications of the developed method is the solution of boundary value problems in areas with axial symmetry [3,11,12].…”