Journal of Optimization, Differential Equations and their Applications (JODEA - Dnipro National Universuty) / Журнал з оптимізації, диференціальних рівнянь та їх застосувань
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PROJECTION-ITERATION REALIZATION OF A NEWTON-LIKE METHOD FOR SOLVING NONLINEAR OPERATOR EQUATIONS
Get PDFWe consider the problem of existence and location of a solution of a nonlinearoperator equation with a Fr´echet differentiable operator in a Banach space and present the convergence results for a projection-iteration method based on a Newton-like method under the Cauchy’s conditions, which generalize the results for the projection-iteration realization of the Newton-Kantorovich method. The proposed method unlike the traditional interpretation is based on the idea of whatever approximation of the original equation by a sequence of approximate operator equations defined on subspaces of the basic space with the subsequent application of the Newton-like method to their approximate solution. We prove the convergence theorem, obtain the error estimate and discuss the advantages of the proposed approach and some of its modifications
ON INITIAL BOUNDARY VALUE PROBLEMS FOR THE DEGENERATE 1D WAVE EQUATION
Get PDFInitial boundary value problems in space-time rectangle for the following linear inhomogeneous degenerate wave equation of the second order smooth coefficient function a(x) vanishes in single points of segment.The well-posedness of the initial boundary value problems is achieved using some approaches to regularization of the equation and the theory of characteristics. The problems for wave equation are then reduced to problems for hyperbolic balance laws 2 and 3 of partial differential equations of the first order. Weak solutions to the problems are obtained using proper numerical methods.Results obtained for some approaches to regularization are presented
ADDENDUM TO ‘A BOLTZMANN CORRECTION’
Get PDFSome implications concerning an unobservable medium whose micro-motion(rotational motion in Faraday vortex tubes) produces magnetic phenomena after Maxwell, used in a previous article on the origin of Maxwell’s equation for the magnetic induction (JODEA, 26 (1), 29 – 44) are clarified and explained
A NEW MATHEMATICAL MODEL OF DYNAMIC PROCESSES IN DIRECT CURRENT TRACTION POWER SUPPLY SYSTEM
Get PDFA new autonomous 4D nonlinear model with two nonlinearities describingthe dynamics of change of voltage and current in the contact railway electric network is offered. This model is a connection of two 2D oscillatory circuits for current and voltage in the contact electric network. In the found system for the defined values of parameters an existence of limit cycles is proved. By introduction of new variables this system can be reduced to 5D system only with one quadratic nonlinearity. The constructed model may be used for the control by voltage stability in a direct current power supply system
A NOTE ON WEIGHTED SOBOLEV SPACES RELATED TO WEAKLY AND STRONGLY DEGENERATE DIFFERENTIAL OPERATORS
Get PDFIn this paper we discuss some issues related to Poincar´e’s inequality for aspecial class of weighted Sobolev spaces. A common feature of these spaces is that they can be naturally associated with differential operators with variable diffusion coefficients that are not uniformly elliptic. We give a classification of these spaces in the 1-D case bases on a measure of degeneracy of the corresponding weight coefficient and study their key properties
A POSSIBILITY OF ROBUST CHAOS EMERGENCE IN LORENZ-LIKE NON-AUTONOMOUS SYSTEM
Get PDFRobust chaos is determined by the absence of periodic windows in bifurcationdiagrams and coexisting attractors with parameter values taken from some regions of the parameter space of a dynamical system. Reliable chaos is an important characteristic of a dynamic system when it comes to its practical application. This property ensures that the chaotic behavior of the system will not deteriorate or be adversely affected by various factors. There are many methods for creating chaotic systems that are generated by adjusting the corresponding system parameters. However, most of the proposed systems are functions of well-known discrete mappings. In view of this, in this paper we consider a continuous system that illustrates some robust chaos properties
ON OPTIMAL 2-D DOMAIN SEGMENTATION PROBLEM VIA PIECEWISE SMOOTH APPROXIMATION OF A SELECTIVE TARGET MAPPING
Get PDFIn this paper we propose a new technique for the solution of the image segmentation problem which is based on the concept of a piecewise smooth approximation of some target functional. We discuss in details the consistency of the new statement of segmentation problem and its solvability. We focus our main intension on the rigor mathematical substantiation of the proposed approach, deriving the corresponding optimality conditions, and show that the new optimization problem is rather flexible and powerful model to the study of variational image segmentation problems. We illustrate the accuracy and efficiency of the proposed algorithm by numerical experiences
AN EXPLICIT SOLVER TO THE DIRICHLET PROBLEM FOR THE LAPLACE EQUATION IN A DISK IN POLYNOMIALS
Get PDFExplicit formulas for the solution to the Dirichlet problem for the Laplaceequation in a disk in polynomials are derived and discussed.Explicit formulas for the solution to the Dirichlet problem for the Laplaceequation in a disk in polynomials are derived and discussed
ON EXISTENCE OF BOUNDED FEASIBLE SOLUTIONS TO NEUMANN BOUNDARY CONTROL PROBLEM FOR p-LAPLACE EQUATION WITH EXPONENTIAL TYPE OF NONLINEARITY
Get PDFWe study an optimal control problem for mixed Dirichlet-Neumann boundary value problem for the strongly non-linear elliptic equation with p-Laplace operator and L1-nonlinearity in its right-hand side. A distribution u acting on a part of boundary of open domain is taken as a boundary control. The optimal control problem is to minimize the discrepancy between a given distribution yd 2 L2( ) and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the state equation for any admissible control. After dening a suitable functional class in which we look for solutions and assuming that this problem admits at least one feasible solution, we prove the existence of optimal pairs. We derive also conditions when the set of feasible solutions has a nonempty intersection with the space of bounded distributions L1( )
On Indirect Approach to the Solvability of Quasi-Linear Dirichlet Elliptic Boundary Value Problem with BMO-Anisotropic p-Laplacian
Get PDFWe study here Dirichlet boundary value problem for a quasi-linear elliptic equation with anisotropic p-Laplace operator in its principle part and L^1-control in coefficient of the low-order term. As characteristic feature of such problem is a specification of the matrix of anisotropy A=A^{sym}+A^{skew} in BMO-space. Since we cannot expect to have a solution of the state equation in the classical Sobolev space W^{1,p}_0(\Omega), we specify a suitable functional class in which we look for solutions and prove existence of weak solutions in the sense of Minty using a non standard approximation procedure and compactness arguments in variable spaces