671 research outputs found
Solution to the Volterra Operator Equations of the 1st kind with Piecewise Continuous Kernels
The sufficient conditions for existence and uniqueness of continuous
solutions of the Volterra operator equations of the first kind with piecewise
continuous kernel are derived. The asymptotic approximation of the parametric
family of solutions are constructed in case of non-unique solution. The
algorithm for the solution's improvement is proposed using the successive
approximations method.Comment: 17 page
Volterra Equations of the First kind with Discontinuous Kernels in the Theory of Evolving Systems Control
The Volterra integral equations of the first kind with piecewise smooth
kernel are considered. Such equations appear in the theory of optimal control
of the evolving systems. The existence theorems are proved. The method for
constructing approximations of parametric families of solutions of such
equations is suggested. The parametric family of solutions is constructed in
terms of a logarithmic-power asymptotics.Comment: 11 pages, bibl 14, Submitted to Studia Informatica Universali
Solution to the Volterra Matrix Equation of the 1st kind with Piecewise Continuous Kernels
In this text matrix Volterra integral equation of the first kind is
addressed. It is assumed that kernels of the equation have jump discontinuities
on non-intersecting curves. Such equations appear in the theory of evolving
dynamic systems. Differentiation of such equations with jump discontinue
kernels yields the new class of the Volterra integral equations with
functionally perturbed argument. The algorithm for construction of the
logarithmic power asymptotics of the desired continuous solutions is proposed.
The theorem of existance of the parametric families of solutions is proved.
Finally the sufficient conditions for existence and uniqueness of continuous
solution are derived.Comment: submitted to Russian Mathematics Journa
Solution to the Volterra integral equations of the first kind with piecewise continuous kernels in class of Sobolev-Schwartz distributions
Sufficient conditions for existence and uniqueness of the solution of the
Volterra integral equations of the first kind with piecewise continuous kernels
are derived in framework of Sobolev-Schwartz distribution theory. The
asymptotic approximation of the parametric family of generalized solutions is
constructed. The method for the solution's regular part refinement is proposed
using the successive approximations method
Existence and Destruction of Kantorovich Main Continuous Solutions of Nonlinear Integral Equations
The sufficient conditions are obtained for existence of the main solution of
the nonlinear Volterra integral equation of the second kind on the semi-axis
and on a finite interval. The method for computation of this boundary interval
is designed. Beyond such integral the solution has the blow-up. The efficiency
of proposed technique is demonstrated on concrete examples
Generalized quadrature for solving singular integral equations of Abel type in application to infrared tomography
We propose the generalized quadrature methods for numerical solution of
singular integral equation of Abel type. We overcome the singularity using the
analytical calculation of the singular integral expression. The problem of
solution of singular integral equation is reduced to nonsingular system of
linear algebraic equations without shift meshes techniques employment. We also
propose generalized quadrature method for solution of Abel equation using the
singular integral. Relaxed errors bounds are derived. In order to improve the
accuracy we use Tikhonov regularization method. We demonstrate the efficiency
of proposed techniques on infrared tomography problem. Numerical experiments
show that it make sense to apply regularization in case of highly noisy sources
only. That is due to the fact that singular integral equations enjoy
selfregularization property
Convex Majorants Method in the Theory of Nonlinear Volterra Equations
The main solutions in sense of Kantorovich of nonlinear Volterra
operator-integral equations are constructed. Convergence of the successive
approximations is established through studies of majorant integral and majorant
algebraic equations. Estimates are derived for the solutions and for the
intervals on the right margin of which the solution has blow-up or solution
start branching
Numerical solution of Volterra integral equations of the first kind with discontinuous kernels
We propose the numerical methods for solution of the weakly regular linear
and nonlinear evolutionary (Volterra) integral equation of the first kind. The
kernels of such equations have jump discontinuities along the continuous curves
(endogenous delays) which starts at the origin. In order to linearize these
equations we use the modified Newton-Kantorovich iterative process. Then for
linear equations we propose two direct quadrature methods based on the
piecewise constant and piecewise linear approximation of the exact solution.
The accuracy of proposed numerical methods is and
respectively. We also suggest a certain iterative
numerical scheme enjoying the regularization properties. Furthermore, we adduce
generalized numerical method for nonlinear equations. We employ the midpoint
quadrature rule in all the cases. In conclusion we include several numerical
examples in order to demonstrate the efficiency of proposed numerical methodsComment: Submitted to APNUM Journa
Numerical Solution of Weakly Regular Volterra Integral Equations of the First Kind
The numerical method for solution of the weakly regular scalar Volterra
integral equation of the 1st kind is proposed. The kernels of such equations
have jump discontinuities on the continuous curves which starts at the origin.
The mid-rectangular quadrature rule is employed. The accuracy of proposed
numerical method is $\mathcal{O}(1/N).
Basins of attraction of nonlinear systems' equilibrium points: stability, branching and blow-up
This paper presents a nonlinear dynamical model which consists the system of
differential and operator equations. Here differential equation contains a
nonlinear operator acting in Banach space, a nonlinear operator equation with
respect to two elements from different Banach spaces. This system is assumed to
enjoy the stationary state (rest points or equilibrium). The Cauchy problem
with the initial condition with respect to one of the desired functions is
formulated. The second function controls the corresponding nonlinear dynamic
process, the initial conditions are not set. The sufficient conditions of the
global classical solution's existence and stabilization at infinity to the rest
point are formulated. It is demonstrated that a solution can be constructed by
the method of successive approximations under the suitable sufficient
conditions. If the conditions of the main theorem are not satisfied, then
several solutions may exist. Some of solutions can blow-up in a finite time,
while others stabilize to a rest point. The special case of considered
dynamical models are nonlinear differential-algebraic equation (DAE) have
successfully modeled various phenomena in circuit analysis, power systems,
chemical process simulations and many other nonlinear processes. Three examples
illustrate the constructed theory and the main theorem. Generalization on the
non-autonomous dynamical systems concludes the article
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