scholarly journals USING OF MULTILAYER NEURAL NETWORKS FOR THE SOLVING SYSTEMS OF DIFFERENTIAL EQUATIONS

2021 ◽  
pp. 81-88
Author(s):  
Natalia Marchenko ◽  
Ganna Sydorenko ◽  
Roman Rudenko
Keyword(s):  
Neural Networks ◽  
Differential Equations ◽  
Loss Function ◽  
Initial Conditions ◽  
Solution Process ◽  
Analytical Form ◽  
Individual Values ◽  
Systems Of Differential Equations ◽  
Solving Equations ◽  
Multilayer Neural Networks

The article considers the study of methods for numerical solution of systems of differential equations using neural networks. To achieve this goal, thefollowing interdependent tasks were solved: an overview of industries that need to solve systems of differential equations, as well as implemented amethod of solving systems of differential equations using neural networks. It is shown that different types of systems of differential equations can besolved by a single method, which requires only the problem of loss function for optimization, which is directly created from differential equations anddoes not require solving equations for the highest derivative. The solution of differential equations’ system using a multilayer neural networks is thefunctions given in analytical form, which can be differentiated or integrated analytically. In the course of this work, an improved form of constructionof a test solution of systems of differential equations was found, which satisfies the initial conditions for construction, but has less impact on thesolution error at a distance from the initial conditions compared to the form of such solution. The way has also been found to modify the calculation ofthe loss function for cases when the solution process stops at the local minimum, which will be caused by the high dependence of the subsequentvalues of the functions on the accuracy of finding the previous values. Among the results, it can be noted that the solution of differential equations’system using artificial neural networks may be more accurate than classical numerical methods for solving differential equations, but usually takesmuch longer to achieve similar results on small problems. The main advantage of using neural networks to solve differential equations` system is thatthe solution is in analytical form and can be found not only for individual values of parameters of equations, but also for all values of parameters in alimited range of values.

On a Numerical Method for Solution of the Mathieu-Hill Type Equations

1980 ◽  
Vol 102 (3) ◽  
pp. 619-626 ◽  
Author(s):  
A. Midha ◽  
M. L. Badlani
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Initial Conditions ◽  
Linear Equations ◽  
Solution Process ◽  
Constraint Equations ◽  
Step Functions ◽  
The Matrix ◽  
Constant Coefficients ◽  
Simultaneous Linear Equations

This paper presents a computer-programmable numerical method for the solution of a class of linear, second order differential equations with periodic coefficients of the Mathieu-Hill type. The method is applicable only when the initial conditions are prescribed and the solution is not requiried to be periodic. The solution is facilitated by representing the coefficient functions as a sum of step functions over corresponding sub-intervals of the fundamental interval. During each sub-interval, the solution form is assumed to be that of the differential equations with “constant” coefficients. Constraint equations are derived from imposing the conditions of “compatibility” of response at the end nodes of the intermediate sub-intervals. This set of simultaneous linear equations is expressed in matrix form. The matrix of coefficients may be represented as a triangular one. This form greatly simplifies the solution process for simultaneous equations. The method is illustrated by its application to some specific problems.

APPLICATION OF INFORMATION TECHNOLOGY FOR MODELING THE CONTROL DYNAMICS OF A NUCLEAR REACTOR ZONING ON THE VERTICAL AXIS

2021 ◽  
Vol 5 ◽  
pp. 45-56
Author(s):  
Valery Severyn ◽  
◽  
Elena Nikulina ◽  
Keyword(s):  
Information Technology ◽  
Nonlinear Systems ◽  
Differential Equations ◽  
Control Systems ◽  
Nuclear Reactor ◽  
Initial Conditions ◽  
Control Object ◽  
Vertical Axis ◽  
Integration Methods ◽  
Systems Of Differential Equations

The structure of information technology for modeling control systems, which includes a block of systems models, a module of integration methods and other program elements, is considered. To analyze the dynamics of control of a nuclear reactor, programs of mathematical models of a WWER-1000 nuclear reactor of the V-320 series and its control systems in the form of nonlinear systems of differential equations in the Cauchy form have been developed. For the integration of nonlinear systems of differential equations, an algorithm of the system method of the first degree is presented. A mathematical model of a WWER-1000 reactor as a control object with division into zones along the vertical axis in relative variables of state is considered, the values of the constant parameters of the model and the initial conditions corresponding to the nominal mode are given. Using information technology for ten zones of the reactor, the system integration method was used to simulate the dynamics of control of a nuclear reactor. Graphs of neutron and thermal processes in the reactor core, as well as changes in the axial offset when the reactor load is dumped under the influence of the movement of absorbing rods and an increase in the concentration of boric acid, are plotted. The analysis of dynamic processes of reactor control is carried out. The programs of integration methods and models of the WWER-1000 reactor of the V-320 series are included in the information technology to optimize the maneuvering modes of the reactor.

SOLUTION OF DIFFERENTIAL EQUATIONS AND SYSTEMS OF DIFFERENTIAL EQUATIONS WITH THE USE OF MULTIPROCESSOR COMPUTERS

2018 ◽  
pp. 59-62 ◽  
Author(s):  
E. V. Glivenko ◽  
A. S. Fomochkina
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Computational Methods ◽  
Initial Conditions ◽  
Geometric Construction ◽  
Cauchy Problems ◽  
Systems Of Differential Equations ◽  
The Cauchy Problem

The paper proposes computational methods for solving differential equations and systems of two differential equations with initial conditions. At the beginning of the article the statement of the problem is given. Then we propose an essential complement to the classical methods for solving the Cauchy problem, which gives a new indication of the correctness of the estimation of the functionsolution. This feature is based on the geometric construction of the solution of several Cauchy problems that differ only in the initial conditions. The article describes an algorithm for constructing a solution on the investigated interval. After that, the possibility of generalizing a feature to systems of two differential equations is described, when the solution is not just a function but a function in space. The methods themselves are easily parallelized, therefore, they can effectively use multiprocessor computers.

scholarly journals Solution of a system of differential equations with constant coefficients using in-verse moments problem techniques

2019 ◽  
Vol 7 (3) ◽  
pp. 71
Author(s):  
Dra. María B. Pintarelli
Keyword(s):  
Differential Equations ◽  
Generating Functions ◽  
Initial Conditions ◽  
Analytical Form ◽  
Inverse Transformation ◽  
General Integral ◽  
Inverse Transform ◽  
Moments Problem ◽  
Constant Coefficients ◽  
Linear Differential

It is known that given a system of simultaneous linear differential equations with constant coefficients you can apply the Laplace method to solve it. The Laplace transforms are found and the problem is reduced to the resolution of an algebraic system of equations of the determining functions, and applying the inverse transformation the generating functions are determined, solutions of the given system. This implies the need to know the analytical form of the inverse transform of the function. In this case the initial conditions consist in knowing the value that the generating function and its derivatives takes at zero. A generalization of this method is proposed in this work, which is to define a more general integral operator than the Laplace transform, the initial conditions consist of Cauchy conditions in the contour. And finally, we find a numerical approximation of the inverse transformation of the generating functions, using the techniques of inverse moment problems, without being necessary to know the analytical form of the inverse transform of the function.

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