scholarly journals Generators of Analytic Resolving Families for Distributed Order Equations and Perturbations

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1306 ◽  
Author(s):  
Vladimir E. Fedorov
Keyword(s):  
Banach Space ◽  
Unbounded Operator ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Perturbation Theorem ◽  
Linear Differential ◽  
Initial Boundary Value ◽  
Families Of Operators ◽  
Lower Order Terms ◽  
Distributed Order

Linear differential equations of a distributed order with an unbounded operator in a Banach space are studied in this paper. A theorem on the generation of analytic resolving families of operators for such equations is proved. It makes it possible to study the unique solvability of inhomogeneous equations. A perturbation theorem for the obtained class of generators is proved. The results of the work are illustrated by an example of an initial boundary value problem for the ultraslow diffusion equation with the lower-order terms with respect to the spatial variable.

scholarly journals Existence results for nonlinear degenerate elliptic equations with lower order terms

2020 ◽  
Vol 10 (1) ◽  
pp. 301-310
Author(s):  
Weilin Zou ◽  
Xinxin Li
Keyword(s):  
Lower Order ◽  
Elliptic Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Regularity Of Solutions ◽  
Degenerate Elliptic Equations ◽  
Degenerate Elliptic ◽  
Initial Boundary Value ◽  
Lower Order Terms ◽  
Nonlinear Degenerate

Abstract In this paper, we prove the existence and regularity of solutions of the homogeneous Dirichlet initial-boundary value problem for a class of degenerate elliptic equations with lower order terms. The results we obtained here, extend some existing ones of [2, 9, 11] in some sense.

scholarly journals Existence of a Unique Weak Solution to a Nonlinear Non-Autonomous Time-Fractional Wave Equation (of Distributed-Order)

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1283
Author(s):  
Karel Van Bockstal
Keyword(s):  
Wave Equation ◽  
Weak Solution ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Order Differential Operator ◽  
Unique Weak Solution ◽  
Fractional Wave Equation ◽  
Equation Of Time ◽  
Initial Boundary Value ◽  
Distributed Order

We study an initial-boundary value problem for a fractional wave equation of time distributed-order with a nonlinear source term. The coefficients of the second order differential operator are dependent on the spatial and time variables. We show the existence of a unique weak solution to the problem under low regularity assumptions on the data, which includes weakly singular solutions in the class of admissible problems. A similar result holds true for the fractional wave equation with Caputo fractional derivative.

scholarly journals EXPLICIT SOLUTION OF THE MIXED PROBLEM IN ANISOTROPIC SEMISPACE WITH BRIGHTLY EXPRESSED VERTICAL PERMEABILITY FOR BARENBLATT–ZHELTOV–KOCHINA EQUATION

2017 ◽  
Vol 18 (6) ◽  
pp. 75-99
Author(s):  
Kh.G. Umarov
Keyword(s):  
Banach Space ◽  
Boundary Value Problem ◽  
Mixed Problem ◽  
Explicit Solution ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Porous Rocks ◽  
Vertical Permeability ◽  
Initial Boundary Value

For Barenblatt–Zheltov–Kochina model presentation of filtering of liquids in fissured-porous rocks the explicit solution of the mixed problem in anisotropic semispace with brightly expressed vertical permeability is found by reduction of the considered problem of filtering to the solution of the abstract initial boundary-value problem in Banach space.

Linear stability analysis in the numerical solution of initial value problems

Acta Numerica ◽  
1993 ◽  
Vol 2 ◽  
pp. 199-237 ◽  
Author(s):  
J.L.M. van Dorsselaer ◽  
J.F.B.M. Kraaijevanger ◽  
M.N. Spijker
Keyword(s):  
Stability Analysis ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Upper Bounds ◽  
New Developments ◽  
Partial Linear ◽  
Linear Differential ◽  
The Stability ◽  
Initial Boundary Value

This article addresses the general problem of establishing upper bounds for the norms of the nth powers of square matrices. The focus is on upper bounds that grow only moderately (or stay constant) where n, or the order of the matrices, increases. The so-called resolvant condition, occuring in the famous Kreiss matrix theorem, is a classical tool for deriving such bounds.Recently the classical upper bounds known to be valid under Kreiss's resolvant condition have been improved. Moreover, generalizations of this resolvant condition have been considered so as to widen the range of applications. The main purpose of this article is to review and extend some of these new developments.The upper bounds for the powers of matrices discussed in this article are intimately connected with the stability analysis of numerical processes for solving initial(-boundary) value problems in ordinary and partial linear differential equations. The article highlights this connection.The article concludes with numerical illustrations in the solution of a simple initial-boundary value problem for a partial differential equation.

scholarly journals On Strongly Continuous Resolving Families of Operators for Fractional Distributed Order Equations

2021 ◽  
Vol 5 (1) ◽  
pp. 20
Author(s):  
Vladimir E. Fedorov ◽  
Nikolay V. Filin
Keyword(s):  
Homogeneous Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Densely Defined Operator ◽  
The Cauchy Problem ◽  
The Laplace Transform ◽  
Initial Boundary Value ◽  
The Right ◽  
Families Of Operators ◽  
Distributed Order

The aim of this work is to find by the methods of the Laplace transform the conditions for the existence of a strongly continuous resolving family of operators for a linear homogeneous equation in a Banach space with the distributed Gerasimov–Caputo fractional derivative and with a closed densely defined operator A in the right-hand side. It is proved that the existence of a resolving family of operators for such equation implies the belonging of the operator A to the class CW(K,a), which is defined here. It is also shown that from the continuity of a resolving family of operators at t=0 the boundedness of A follows. The existence of a resolving family is shown for A∈CW(K,a) and for the upper limit of the integration in the distributed derivative not greater than 2. As corollary, we obtain the existence of a unique solution for the Cauchy problem to the equation of such class. These results are used for the investigation of the initial boundary value problems unique solvability for a class of partial differential equations of the distributed order with respect to time.

A Convective Description of Flexible Robot Dynamics

2002 ◽  
Author(s):  
Daniel Franitza
Keyword(s):  
Differential Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Dynamics Simulation ◽  
Robot Dynamics ◽  
Static Case ◽  
Flexible Robot ◽  
Non Linear ◽  
Linear Differential ◽  
Initial Boundary Value

This paper describes a theory for dynamics simulation of non-linear elastically deformable mechanisms as a basis for compliant manipulators. The description of the mechanical structure based on local coordinates in a convective metric for one-dimensional structures is introduced. Besides the quasi-static case, loads related to mass inertial terms are taken into consideration. This changes the original first order set of linear differential equations into non-linear second order partial differential equations. Some numerical aspects for the generation of a solution for the initial boundary value problem described are discussed.

Existence and Asymptotic Behavior for a Strongly Damped Nonlinear Wave Equation

1980 ◽  
Vol 32 (3) ◽  
pp. 631-643 ◽  
Author(s):  
G. F. Webb
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Ordinary Differential Equation ◽  
Asymptotic Behavior ◽  
Global Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Smooth Bounded Domain ◽  
Initial Boundary Value ◽  
Strongly Damped

In this paper we study the nonlinear initial boundary value problem(1.1) ωtt— αΔ ωt— Δω= f(ω), t> 0ω(x, 0) = ϕ(x), x∈ Ωωt(x, 0) = ψ (x), x∈ Ωω(x, t ) = 0, x ∈ ∂Ω, t ≥ 0.In (1.1) Ω is a smooth bounded domain in Rn, n = 1, 2, 3, α > 0, and f ∈ C1(R;R) with f‘(x) ≦ co for all x ∈ R (where c0 is a nonnegative constant), lim sup|x|→+∞f(x)/x ≦0, and f(0) = 0. Our objective will be to establish the existence of unique strong global solutions to (1.1) and investigate their behavior as t→ +∞.Our approach takes advantage of the semilinear character of (1.1) and reformulates the problem as an abstract ordinary differential equation in a Banach space.

Convergence of Finite Fifference Method for Parabolic Problem

2003 ◽  
Vol 3 (1) ◽  
pp. 45-58 ◽  
Author(s):  
Dejan Bojović
Keyword(s):  
Convergence Rate ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Parabolic Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variable Coefficients ◽  
Rate Estimate ◽  
Convergence Rate Estimate ◽  
Initial Boundary Value

Abstract In this paper we consider the first initial boundary-value problem for the heat equation with variable coefficients in a domain (0; 1)x(0; 1)x(0; T]. We assume that the solution of the problem and the coefficients of the equation belong to the corresponding anisotropic Sobolev spaces. Convergence rate estimate which is consistent with the smoothness of the data is obtained.

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