scholarly journals Optimal time and space regularity for solutions of degenerate differential equations

2009 ◽  
Vol 7 (2) ◽  
Author(s):  
Alberto Favaron
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Cauchy Problem ◽  
Differential Equations ◽  
Optimal Time ◽  
Time And Space ◽  
Optimal Regularity ◽  
Degenerate Differential Equation ◽  
The Cauchy Problem ◽  
Degenerate Differential Equations

AbstractWe derive optimal regularity, in both time and space, for solutions of the Cauchy problem related to a degenerate differential equation in a Banach space X. Our results exhibit a sort of prevalence for space regularity, in the sense that the higher is the order of regularity with respect to space, the lower is the corresponding order of regularity with respect to time.

ON THE CAUCHY PROBLEM FOR SOME PSEUDO-DIFFERENTIAL EQUATIONS OF SCHRÖDINGER TYPE

1996 ◽  
Vol 06 (03) ◽  
pp. 295-314 ◽  
Author(s):  
R. AGLIARDI ◽  
D. MARI
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Fundamental Solution ◽  
Differential Equations ◽  
Lower Order ◽  
Order Term ◽  
Well Posedness ◽  
Gevrey Spaces ◽  
The Cauchy Problem ◽  
Pseudo Differential Equation

A fundamental solution of the Cauchy problem is constructed for a pseudo-differential equation generalizing some Schrödinger equations. Then well-posedness of the Cauchy problem is proved in some Gevrey spaces whose indices depend on the lower order term of the operator.

scholarly journals On the exact and approximate solutions of several differential equations with variational derivatives of the first and second orders

2020 ◽  
Vol 56 (1) ◽  
pp. 51-71
Author(s):  
M. V. Ignatenko ◽  
L. A. Yanovich
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Approximate Solution ◽  
Differential Equations ◽  
Interpolation Problem ◽  
Approximate Solutions ◽  
Hyperbolic Type ◽  
The Cauchy Problem ◽  
Variational Derivatives ◽  
Derivatives Of

In this paper, we consider the problem of the exact and approximate solutions of certain differential equations with variational derivatives of the first and second orders. Some information about the variational derivatives and explicit formulas for the exact solutions of the simplest equations with the first variational derivatives are given. An interpolation method for solving ordinary differential equations with variational derivatives is demonstrated. The general scheme of an approximate solution of the Cauchy problem for nonlinear differential equations with variational derivatives of the first order, based on the use of the operator interpolation apparatus, is presented. The exact solution of the differential equation of the hyperbolic type with variational derivatives, similar to the classical Dalamber solution, is obtained. The Hermite interpolation problem with the conditions of coincidence in the nodes of the interpolated and interpolation functionals, as well as their variational derivatives of the first and second orders, is considered for functionals defined on the sets of differentiable functions. The found explicit representation of the solution of the given interpolation problem is based on an arbitrary Chebyshev system of functions. This solution is generalized for the case of interpolation of functionals on one out of two variables and applied to construct an approximate solution of the Cauchy problem for the differential equation of the hyperbolic type with variational derivatives. The description of the material is illustrated by numerous examples.

On the existence of a continuously differentiable solution to the Cauchy problem for implicit differential equations

2019 ◽  
pp. 376-383
Author(s):  
Zukhra T. Zhukovskaya ◽  
Sergey E. Zhukovskiy
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Differential Equations ◽  
Initial Point ◽  
Local Solvability ◽  
Time Interval ◽  
Phase Variable ◽  
Existence Of A Solution ◽  
Differentiable Functions ◽  
The Cauchy Problem

We study the question of the existence of a solution to the Cauchy problem for a differential equation unsolved with respect to the derivative of the unknown function. Differential equations generated by twice continuously differentiable mappings are considered. We give an example showing that the assumption of regularity of the mapping at each point of the domain is not enough for the solvability of the Cauchy problem. The concept of uniform regularity for the considered mappings is introduced. It is shown that the assumption of uniform regularity is sufficient for the local solvability of the Cauchy problem for any initial point in the class of continuously differentiable functions. It is shown that if the mapping defining the differential equation is majorized by mappings of a special form, then the solution of the Cauchy problem under consideration can be extended to a given time interval. The case of the Lipschitz dependence of the mapping defining the equation on the phase variable is considered. For this case, estimates of non-extendable solutions of the Cauchy problem are found. The results are compared with known ones. It is shown that under the assumptions of the proved existence theorem, the uniqueness of a solution may fail to hold. We provide examples llustrating the importance of the assumption of uniform regularity.

scholarly journals ON THE ALGORITHM FOR SOLVING OF A LINEAR BOUNDARY VALUE PROBLEM FOR ORDINARY DIFFERENTIAL EQUATION WITH PARAMETER

2020 ◽  
Vol 70 (2) ◽  
pp. 71-76
Author(s):  
N.B. Iskakova ◽  
◽  
Zh. Kubanychbekkyzy ◽  
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Ordinary Differential Equations ◽  
Boundary Value ◽  
Linear Boundary ◽  
Existence Of A Solution ◽  
The Cauchy Problem ◽  
Linear Boundary Value Problem

A linear boundary value problem for a system of ordinary differential equations containing a parameter is considered on a bounded segment. For a fixed parameter value, the Cauchy problem for an ordinary differential equation is solved. Using the fundamental matrix of differential part and assuming uniqueness solvability of the Cauchy problem an origin boundary value problem is reduced to the system of linear algebraic equation with respect to unknown parameter. The existence of a solution to this system ensures the existence of a solution to the boundary value problem under study. The algorithm of finding of solution for initial problem is offered based on a construction and solving of the linear algebraic equation. The basic auxiliary problem of algorithm is: the Cauchy problem for ordinary differential equations. The numerical implementation of algorithm offered in the article uses the method of Runge-Kutta of fourth order to solve the Cauchy problem for ordinary differential equations.

scholarly journals Asymptotic Solutions of Scalar Integro-Differential Equations with Partial Derivatives and with Fast Oscillating Coefficients

2020 ◽  
Vol 13 (2) ◽  
pp. 287-302
Author(s):  
Burkhan Kalimbetov ◽  
Akisher Temirbekov ◽  
Abdimuhan Tolep
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Differential Equations ◽  
Unique Solvability ◽  
Regularization Method ◽  
Singularly Perturbed ◽  
Integral Term ◽  
Integro Differential Equation ◽  
The Cauchy Problem ◽  
Oscillating Coefficients

In the paper, ideas of the Lomov regularization method are generalized to the Cauchy problem for a singularly perturbed partial integro-differential equation in the case when the integral term contains a rapidly varying kernel. Regularization of the problem is carried out, the normal and unique solvability of general iterative problems is proved.

scholarly journals DENSENESS OF SETS OF CAUCHY PROBLEMS WITHHOUT SOLUTIONS AND WITH NONUNIQUE SOLUTIONS IN THE SET OF ALL CAUCHY PROBLEMS

2020 ◽  
Vol 8 (2) ◽  
pp. 122-126
Author(s):  
V. Slyusarchuk
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Differential Equations ◽  
Cauchy Problems ◽  
Number Of Solutions ◽  
Computational Mathematics ◽  
Infinite Dimensional ◽  
Infinite Dimensional Banach Space ◽  
The Cauchy Problem ◽  
Solutions Of Equations

When finding solutions of differential equations it is necessary to take into account the theorems on innovation and unity of solutions of equations. In case of non-fulfillment of the conditions of these theorems, the methods of finding solutions of the studied equations used in computational mathematics may give erroneous results. It should also be borne in mind that the Cauchy problem for differential equations may have no solutions or have an infinite number of solutions. The author presents two statements obtained by the author about the denseness of sets of the Cauchy problem without solutions (in the case of infinite-dimensional Banach space) and with many solutions (in the case of an arbitrary Banach space) in the set of all Cauchy problems. Using two examples of the Cauchy problem for differential equations, the imperfection of some methods of computational mathematics for finding solutions of the studied equations is shown.

ASYMPTOTIC SOLUTION OF FIRST-ORDER EQUATION WITH SMALL PARAMETER UNDER THE DERIVATIVE WITH PERTURBED OPERATOR

2018 ◽  
pp. 784-796
Author(s):  
Vladimir Igorevich Uskov
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Cauchy Problem ◽  
Asymptotic Expansion ◽  
Asymptotic Solution ◽  
Small Parameter ◽  
Decomposition Method ◽  
Order Equation ◽  
First Order ◽  
The Cauchy Problem

The paper is devoted to the Cauchy problem for a differential equation with a small parameter when using a Fredholm operator in a Banach space with a certain method. The investigated effect of this parameter. The solution is in the form of an asymptotic expansion. When solving the problems of using the cascade decomposition method for equations, which allows us to split the equation into equations in subspaces.

Sign in / Sign up

Export Citation Format

Share Document