A generalization of the method of separation of variables

Kalenyuk, P. І.

doi:10.1007/bf01085281

Cited by 3 publications

(10 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The existence and uniqueness class for the solution (6) is the class of functions that have a single-valued analytic continuation to entire functions of order less than 2 for each t > 0 [9]. Formulas (3)-(6), as we see, are applicable for different classes of initial functions.…”

Section: The Heat Equationmentioning

confidence: 97%

“…In order for a solution (5) of the problem (2) to exist the funqtion rp must belong to the Sobolev space of infinite order [7] Applying the operational method of generalized separation of variables to the problem (2) makes it possible to obtain the following formal solution [9]:…”

Section: The Heat Equationmentioning

confidence: 99%

“…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.…”

mentioning

confidence: 99%

See 2 more Smart Citations

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

Kalenyuk¹,

Nytrebych²

1999

J Math Sci

Self Cite

View full text Add to dashboard Cite

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...

show abstract

Section: The Heat Equationmentioning

confidence: 97%

Section: The Heat Equationmentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

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

Kalenyuk¹,

Nytrebych²

1999

J Math Sci

Self Cite

View full text Add to dashboard Cite

show abstract

“…The problem (9), (10) has been studied in the work [11] by means of the differential-symbol method [12,13]. We shall represent this problem as problem (1), (2), in which A = d dx , exp[λx] is an eigenfunction of the operator A, H is a class of entire functions, L = K M is a class of quasipolynomials…”

Section: Problem With Integral Condition For Partial Differential Equmentioning

confidence: 99%

Problem with integral condition for evolution equation

Kalenyuk¹,

Kuduk²,

Kohut³

et al. 2015

JMA

View full text Add to dashboard Cite

Abstract:We propose a method of solving the problem with nonhomogeneous integral condition for homogeneous evolution equation with abstract operator in a linear space H. For right-hand side of the integral condition which belongs to the special subspace L ⊆ H, in which the vectors are represented using Stieltjes integrals over a certain measure, the solution of the problem is represented in the form of Stieltjes integral over the same measure.

show abstract

“…On the basis of a generalized separation-of-variables method [4] an operator method of constructing the solution of the Cauchy problem has been proposed [8,9,11] for a general partial differential equation solved with respect to the highest order time derivative and constant or time-variable coefficients..…”

mentioning

confidence: 99%

An operator method of solving the Cauchy problem for a homogeneous system of partial differential equations

Нитребич¹

1996

J Math Sci

View full text Add to dashboard Cite

517.944On the basis of a generalized separation-of-variables method we propose an operator method of constructing the solution of the Cauchy problem for a homogeneous system of partial differential equations of first order with respect to time an d of infinite order with respect to the spatial variables.It is known that the classical operator (symbolic) calculus is a method of solving differential equations in which the differentiation symbol p = d/dx operates as an ordinary variable. Such a method, proposed by M. Vashchenko-Zakharchenko and subsequently developed in the work of O. Heaviside, yielded a number of important results in its time. Later this method was given a foundation using the Laplace transform.The extension of the classical operational calculus to partial differential equations and systems of such equations for specific problems of mechanics is connected with the work of A. I. Lur'e [7]. In particular, to solve problems of elasticity theory Lur'e developed an operator method of finding a solution of, for example, the Cauchy problem, in the form of an action applied to the initial functions of certain differential operators, of generally infinite order with respect to the spatial variables.On the basis of a generalized separation-of-variables method [4] an operator method of constructing the solution of the Cauchy problem has been proposed [8, 9, 11] for a general partial differential equation solved with respect to the highest order time derivative and constant or time-variable coefficients..The present paper is devoted to the construction of an analogous method of solving the Cauchy problem for a homogeneous system of partial differential equations and constant coefficients.In the variable space t > 0, x = (xl,x2,... ,xs) E R s we study the Cauchy problemwhere u (~, x) = (u~(t, ~), u2(t, ~),..., u~(t, x)

show abstract

A generalization of the method of separation of variables

Cited by 3 publications

References 1 publication

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

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

Problem with integral condition for evolution equation

An operator method of solving the Cauchy problem for a homogeneous system of partial differential equations

Contact Info

Product

Resources

About