Uniqueness of solutions of a mixed problem for parabolic differential-functional equations

Szarski, J.

doi:10.4064/ap-28-1-57-65

Cited by 12 publications

(8 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let (x*,y*) E T. We have u(x,y) < v(x,y) for x < x* and (x,y) 9 E, and u(x*,y*) --v(x*,y*). From (15) and from assumption 2 0 we have In both case we get a contradiction with (16) and it proves that u(x, y) < v(x, y) for (x,v) eE*. 9 exists on [-h, a) and ~imw(t, e) = 0 uniformly on [-h, a).…”

Section: U) -~(~0 + T Y)l)t-h A))mentioning

confidence: 87%

See 1 more Smart Citation

Differential-functional inequalities related to initial-boundary problems for first order partial differential-functional equations

Jaruszewska-Walczak

1993

Period Math Hung

View full text Add to dashboard Cite

The paper deals with the theory of differential-functional inequalities and with its applications to mixed problems for first order partial differential-functional equations with initial-boundary conditions.The theory of partial differential inequalities was originated by Haar [5] and Nagumo [13]. The classical theory of partial differential inequalities is in detail described in monographs [9], [15]. The main applications of the theory concern questions such as: estimates of solutions of partial equations, estimates of the difference between solutions of two initial problems, estimates of the existence domain of solutions, criteria of the uniqueness of the solutions, stability criteria, estimates of the error for an approximate solution, continuous dependence of the solution on initial data and on the right hand side of the system. A similar role in the theory of differential-functional equations with the first order partial derivatives is played by ordinary differential-functional inequalities. Some results in this field can be found in [4], [6], [7], [14]. All results in the papers cited above concern initial value problems for first order partial differential-functional equations.In this paper we consider mixed initial-boundary value problems for differential-functional equations. We prove a theorem on the uniqueness of solutions of Perron type. We obtain this uniqueness theorem as a particular case of some general comparison theorem for differential-functional inequalities with initial-boundary conditions.In the last part of the paper we give a generalization of the well-known theorems on differential or differential-functional inequalities with initial conditions ([9], [14], [15]) on the case of differential-functional inequalities with initial-boundary conditions. Let C(X, Y) denote the class of all continuous mappings from X into Y where X, Y are metric spaces. We shall use vectorial inequalities if the same inequalities hold between their corresponding components. We shall denote a function w of the variable t 9 [c~,fl) C R by (w(t))[a,~). Denote by 11. IIc the supremum norm in theWe introduce the following assumption (see [16]). Assumption H0. Suppose that a region f~ C R l+n satisfies the following conditions:Mathematics subject classification numbers, 1991. Primary 34A40, 34A99, 34B99.

show abstract

Section: U) -~(~0 + T Y)l)t-h A))mentioning

confidence: 87%

“…Denote by 11. IIc the supremum norm in theWe introduce the following assumption (see [16]). Assumption H0.…”

mentioning

confidence: 99%

Differential-functional inequalities related to initial-boundary problems for first order partial differential-functional equations

Jaruszewska-Walczak

1993

Period Math Hung

View full text Add to dashboard Cite

show abstract

“…The existence of the solutions is studied in [4]. The uniqueness of the solutions is studied in [7].…”

Section: Parabolic Functional Differential Equationsmentioning

confidence: 99%

Error bounds for spatial discretization and waveform relaxation applied to parabolic functional differential equations

Zubik‐Kowal

2004

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

The process of semi-discretization and waveform relaxation are applied to general nonlinear parabolic functional differential equations. Two new theorems are presented, which extend and improve some of the classical results. The first of these theorems gives an upper bound for the norm of the error of finite difference semi-discretization. This upper bound is sharper than the classical error bound. The second of these theorems gives an upper bound for the norm of the error, which is caused by both semi-discretization and waveform relaxation. The focus in the paper is on estimating this error directly without using the upper bound for the error, which is caused by the process of semidiscretization and the upper bound for the error, which is caused by the waveform relaxation method. Such estimating gives sharper error bound than the bound, which is obtained by estimating both errors separately.  2004 Elsevier Inc. All rights reserved.

show abstract

“…Sufficient conditions for the existence and uniqueness of classical or generalized solutions of parabolic functional differential problems can be found in [2,6,17]. Functional differential inequalities and applications were studied in [16,19]. The monographs [8,22] give the exposition of the theory of partial functional differential equations.…”

Section: Introductionmentioning

confidence: 99%

Implicit difference methods for evolution functional differential equations

Kamont

Kropielnicka

2011

Numer. Analys. Appl.

View full text Add to dashboard Cite

A general theory of implicit difference schemes for nonlinear functional differential equations with initial boundary conditions is presented. A theorem on error estimates of approximate solutions for implicit functional difference equations of the Volterra type with an unknown function of several variables is given. This general result is employed to investigate the stability of implicit difference schemes generated by first-order partial differential functional equations and by parabolic problems. A comparison technique with nonlinear estimates of the Perron type for given functions with respect to the functional variable is used.

show abstract

Uniqueness of solutions of a mixed problem for parabolic differential-functional equations

Cited by 12 publications

References 0 publications

Differential-functional inequalities related to initial-boundary problems for first order partial differential-functional equations

Differential-functional inequalities related to initial-boundary problems for first order partial differential-functional equations

Error bounds for spatial discretization and waveform relaxation applied to parabolic functional differential equations

Implicit difference methods for evolution functional differential equations

Contact Info

Product

Resources

About