On an operational method of solving initial-value problems for partial differential equations induced by generalized separation of variables

Abstract

517.95We propose an operational method of solving the Cauchy problem for partial differential equations and systems of partial differential equations. We demonstrate its superiority to the known methods. We give a number of illustrative examples of applications of the method.The efforts of many scholars have amassed a large number of methods of solving both partial differential equations and initial-value problems connected with them. Among such methods are the well-known operator method [ 1,7,16], the method of integral transforms [22], the method of initial functions [6] and others. The application of each such method to a specific Cauchy problem, wherever this can be done, makes it possible to obtain different presentations of its solution. To the methods of solving the Cauchy problem mentioned above one may add the operational method [9-11, 13, 19], which is based on generalized separation of variables [9,12,14]. This method is in a certain sense "dual" to the operator method whose starting point is the classical operational calculus.The classical operational calculus is a method of solving ordinary differential equations in which the symbol p = d/dx is manipulated like an ordinary variable. For the required solution one obtains expressions of the formwhere f(x) is a known function. Such a calculus was first proposed in [4], and then greatly developed in [25]. The basic problem in this calculus is obviously to find an effective interpretation of the symbols g(p). We note that a complete mathematical justification of the symbolic calculus was not given in these papers. The justification was given much later, when a connection was established between this calculus and the Laplace transform. What is important in this topic was the algebraic approach to justifying the operational calculus proposed in [17]. The technique of infinite-order differential operators which is inherent in the classical symbolic calculus, has recently been successfully applied to study Cauchy problems for partial differential equations (see, for example, [7,8] and the bibliography they contain).The formalism of infinite-order differential operators has frequently been applied in recent years to study specific problems of mechanics [3,6,15,20]. In these papers the solutions of initial-and boundary-value problems are given using the expressions (1) where x e ~", p = (Pl , P2 ..... P,), Pj = 6~/~ , J = 1, n.The operational method of generalized separation of variables considered in the present paper makes it possible to solve the Cauchy problem using expressions of the form where fj are known functions (initial functions or right-hand sides of the equations), and # e C'.The proposed operational method of generalized separation of variables has certain advantages over those mentioned above, and the present paper is devoted to a description of those advantages. The characteristics of the method can be illustrated using the examples of Cauchy problems for the equations of the direct and inverse heatconduction problems, wave equations for the ca...

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