scholarly journals Analysis of approaches to mass-transfer modeling n non-stationary mode

2019 ◽  
pp. 55-64
Author(s):  
Yaroslav Pyanylo ◽  
Galyna Pyanylo
Keyword(s):  
Differential Equation ◽  
Mass Transfer ◽  
Numerical Methods ◽  
Differential Equations ◽  
Stationary Mode ◽  
System Of Equations ◽  
Physical Processes ◽  
Partial Derivatives ◽  
Systems Of Differential Equations ◽  
Mass Transfer Modeling

A significant number of natural and physical processes are described by differential equations in partial derivatives or systems of differential equations in partial derivatives. Numerical methods have been found to find their solutions. Partial derivatives systems are solved mainly by reducing the order of the system of equations or reducing it to one differential equation. This procedure leads to an increase in the order of the differential equation. There are various restrictions and errors that can lead to additional solutions, boundary conditions for intermediate derivatives, and so on. The work is devoted to the analysis of such situations and ways of exit.

scholarly journals On the Question of Asymptotic Integration of Singularly Perturbed Fractional-Order Problems

2018 ◽  
Vol 6 (3) ◽  
Author(s):  
Burkhan Kalimbetov
Keyword(s):  
Differential Equations ◽  
Small Parameter ◽  
Fractional Order ◽  
Initial Problem ◽  
Singularly Perturbed ◽  
Asymptotic Integration ◽  
Partial Derivatives ◽  
Systems Of Differential Equations ◽  
Regularization Problem ◽  
Iterative Systems

In this paper we consider an initial problem for systems of differential equations of fractional order with a small parameter for the derivative. Regularization problem is produced, and algorithm for normal and unique solubility of general iterative systems of differential equations with partial derivatives is given. 

scholarly journals A Note on Uniqueness and Convergence of Successive Approximations

1959 ◽  
Vol 2 (1) ◽  
pp. 5-8 ◽  
Author(s):  
Fred Brauer
Keyword(s):  
Differential Equation ◽  
General Theory ◽  
Differential Equations ◽  
Successive Approximations ◽  
Uniqueness Of Solutions ◽  
Convergence Theorems ◽  
Main Interest ◽  
Systems Of Differential Equations ◽  
Two Stages

In a recent note in this Bulletin [3], W.A.J. Luxemburg has shown in two different ways that a condition of Krein and Krasnosel'skii [2] for the uniqueness of solutions of a differential equation also implies the convergence of the successive approximations. Here, a third proof of the uniqueness and the convergence of successive approximations, formulated for systems of differential equations, will be obtained. This third proof is modelled on the methods used in proving general uniqueness and convergence theorems [l]. The approach is suggested by Luxemburg's idea of breaking the argument into two stages and using one of the hypotheses in each stage. Since the proofs given here are hardly shorter than the earlier direct proofs, their main interest lies in the fact that they fit what appeared to be an isolated result into the framework of a general theory.

scholarly journals On time transformations for differential equations with state-dependent delay

2014 ◽  
Vol 12 (2) ◽  
Author(s):  
Alexander Rezounenko
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Asymptotic Properties ◽  
The State ◽  
Delay Systems ◽  
Constant Delay ◽  
Systems Of Differential Equations ◽  
State Dependent ◽  
State Dependent Delay ◽  
Time Transformations

AbstractSystems of differential equations with state-dependent delay are considered. The delay dynamically depends on the state, i.e. is governed by an additional differential equation. By applying the time transformations we arrive to constant delay systems and compare the asymptotic properties of the original and transformed systems.

scholarly journals SIMULATION OF OSCILLATORY PROCESSES USING DIFFERENTIAL EQUATIONS OF LIOUVILLE TYPE

2018 ◽  
pp. 117-123
Author(s):  
Albina Kuandykovna Ilyasova ◽  
Yuliia Vladimirovna Bulycheva
Keyword(s):  
Mathematical Modeling ◽  
Differential Equation ◽  
Differential Equations ◽  
Explicit Form ◽  
Physical Phenomenon ◽  
Integral Representations ◽  
Explicit Solutions ◽  
Cauchy Type ◽  
Partial Derivatives ◽  
Oscillatory Processes

The problems of mathematical modeling lead to the necessity to create computational algorithms directly related to finding solutions of differential equations with partial derivatives in explicit form. In this study, explicit solutions are original tests for approximate methods that reflect the essence of the general solution. Each explicit solution of the differential equation has great importance as an accurate representation of the physical phenomenon under study within the framework of this model, as an analysis of the verification of numerical methods, as a theoretical basis for further modeling of the researched process. There have been considered aspects of the application of mathematical modeling to the study of oscillatory processes. Methods of reducing the solution of differential equations to an explicit form are proposed. Solution is given through functions of real arguments. The possible field of application is the study of wave processes. There is being considered the problem of building a variety of explicit solutions of the nonlinear third-order differential equation with partial derivatives with two boundary singular planes in space and second-order equation of general form with hyper-singular lines in the plane. On the basis of the developed method there has been proved the uniqueness of the obtained integral representations, and the boundary value problem of Cauchy type is posed and solved. The results are formulated in the form of theorems.

scholarly journals Methods for solving the initial value problem with a two-sided estimate of the local error

2021 ◽  
pp. 88-92
Author(s):  
Yaroslav Pelekh ◽  
Andrii Kunynets ◽  
Halyna Beregova ◽  
Tatiana Magerovska
Keyword(s):  
Differential Equation ◽  
Numerical Methods ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problem ◽  
Local Error ◽  
Initial Value ◽  
Order Of Accuracy ◽  
Embedded Methods ◽  
The Right

Numerical methods for solving the initial value problem for ordinary differential equations are proposed. Embedded methods of order of accuracy 2(1), 3(2) and 4(3) are constructed. To estimate the local error, two-sided calculation formulas were used, which give estimates of the main terms of the error without additional calculations of the right-hand side of the differential equation, which favorably distinguishes them from traditional two-sided methods of the Runge- Kutta type.

scholarly journals Analytical Solutions of Fuzzy Linear Differential Equations in the Conformable Setting

2021 ◽  
Vol 2 (2) ◽  
pp. 13-30
Author(s):  
Awais Younus ◽  
Muhammad Asif ◽  
Usama Atta ◽  
Tehmina Bashir ◽  
Thabet Abdeljawad
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Analytical Solutions ◽  
Higher Order ◽  
Linear Differential Equations ◽  
Conformable Derivative ◽  
Systems Of Differential Equations ◽  
Two Systems ◽  
Linear Differential

In this paper, we provide the generalization of two predefined concepts under the name fuzzy conformable differential equations. We solve the fuzzy conformable ordinary differential equations under the strongly generalized conformable derivative. For the order $\Psi$, we use two methods. The first technique is to resolve a fuzzy conformable differential equation into two systems of differential equations according to the two types of derivatives. The second method solves fuzzy conformable differential equations of order $\Psi$ by a variation of the constant formula. Moreover, we generalize our results to solve fuzzy conformable ordinary differential equations of a higher order. Further, we provide some examples in each section for the sake of demonstration of our results.

scholarly journals IV. On linear differential equations

1870 ◽  
Vol 18 (114-122) ◽  
pp. 118-119
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Rational Function ◽  
Linear Differential Equations ◽  
System Of Equations ◽  
Linear Differential

The condition that the linear differential equation (α + β x + γ x 2 ) d 2 u / dx 2 + (α' + β' x + γ' x 2 ) du / dx + (α'' + β" x + γ" x 2 ) u = 0 admits of an integral u = ϵ fφdx , where φ is a rational function of ( x ), is given by the system of equations

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