The method of successive approximations and a uniqueness-theorem of krasnoselskil and krein in the theory of differential equations

Uniqueness Of Solutions ◽

Picard Method

We consider the random fuzzy differential equations (RFDEs) with impulses. Using Picard method of successive approximations, we shall prove the existence and uniqueness of solutions to RFDEs with impulses under suitable conditions. Some of the properties of solution of RFDEs with impulses are studied. Finally, an example is presented to illustrate the results.

METHODS FOR SOLVING FREDHOLM INTEGRAL EQUATIONS

Bulletin Series of Physics & Mathematical Sciences ◽

10.51889/2020-1.1728-7901.29 ◽

2020 ◽

Vol 69 (1) ◽

pp. 174-178

Author(s):

R.S. Ysmagul ◽

◽

А.Е. Nurgeldina ◽

Keyword(s):

Quantum Mechanics ◽

Initial Data ◽

Differential Equations ◽

Integral Equations ◽

Fredholm Integral Equations ◽

Future State ◽

Continuous Sequence ◽

The Moment

The article deals with integral equations that are widely used in various sections of physics (theory of waves on the surface of liquids, quantum mechanics, problems of spectroscopy, crystallography, acoustics, analysis and diagnostics of plasma, etc.), Geophysics (problems of gravimetry, kinematic problems of seismics), mechanics (vibrations of structures), etc. When the physics introduced aftereffect, it is not enough ordinary differential equations or partial differential equations, otherwise the initial data would determine the future state. To take into account the continuous sequence of previous States, we need to use integral and integro-differential equations, where the sign of the integral appears functions of parameters that characterize the system, which depend on time for some period preceding the moment under consideration. In this article we have considered the solution of Fredholm integral equations of the second kind by the method of successive approximations and the method of iterated nuclei.

First Order Partial Functional Differential Equations with Unbounded Delay

Georgian Mathematical Journal ◽

10.1515/gmj.2003.509 ◽

2003 ◽

Vol 10 (3) ◽

pp. 509-530

Author(s):

Z. Kamont ◽

S. Kozieł

Keyword(s):

Differential Equations ◽

Integral Functional ◽

Functional Differential Equations ◽

Generalized Solutions ◽

First Order ◽

Unbounded Delay ◽

The Cauchy Problem ◽

Functional Differential

Abstract The phase space for nonlinear hyperbolic functional differential equations with unbounded delay is constructed. The set of axioms for generalized solutions of initial problems is presented. A theorem on the existence and continuous dependence upon initial data is given. The Cauchy problem is transformed into a system of integral functional equations. The existence of solutions of this system is proved by the method of successive approximations and by using theorems on integral inequalities. Examples of phase spaces are given.

About the existence and uniqueness theorem for hyperbolic equation

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171295000184 ◽

1995 ◽

Vol 18 (1) ◽

pp. 141-146

Author(s):

M. E. Khalifa

Keyword(s):

Hyperbolic Equation ◽

Uniqueness Theorem ◽

Existence And Uniqueness ◽

Existence And Uniqueness Theorem ◽

Almost Everywhere

In this paper we prove the existence and uniqueness theorem for almost everywhere solution of the hyperbolic equation using the method of successive approximations [1].

Solving Some Derivative Equations Fractional Order Nonlinear Partials Using the Some Blaise Abbo Method

Journal of Mathematics Research ◽

10.5539/jmr.v13n2p101 ◽

2021 ◽

Vol 13 (2) ◽

pp. 101

Author(s):

Abdoul wassiha NEBIE ◽

Frederic BERE ◽

Bakari ABBO ◽

Youssouf PARE

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Fractional Order ◽

Decomposition Method ◽

Nonlinear Partial Differential Equations ◽

Adomian Decomposition Method ◽

Adomian Decomposition ◽

Diffusion Convection

In this paper, we propose the solution of some nonlinear partial differential equations of  fractional order that modeled diffusion, convection and reaction problems. For the solution of these equations we will use the SBA method which is a method based on the combination of the Adomian Decomposition Method (ADM), the Picard's principle  and the method of successive approximations.   

On solvability of one class of third order differential equations

Ukrains’kyi Matematychnyi Zhurnal ◽

10.37863/umzh.v73i3.195 ◽

2021 ◽

Vol 73 (3) ◽

pp. 314-328

Author(s):

B. T. Bilalov ◽

M. I. Ismailov ◽

Z. A. Kasumov

Keyword(s):

Differential Equations ◽

Fourier Method ◽

Generalized Solution ◽

Countable System ◽

Order Partial Differential Equation ◽

Third Order ◽

Right Hand ◽

The Right

UDC 517.9 One-dimensional mixed problem for one class of third order partial differential equation with nonlinear right-hand side is considered. The concept of generalized solution for this problem is introduced. By the Fourier method, the problem of existence and uniqueness of generalized solution for this problem is reduced to the problem of solvability of the countable system of nonlinear integro-differential equations. Using Bellman's inequality, the uniqueness of generalized solution is proved. Under some conditions on initial functions and the right-hand side of the equation, the existence theorem for the generalized solution is proved using the method of successive approximations.

Some Results on Uniqueness and Successive Approximations

Canadian Journal of Mathematics ◽

10.4153/cjm-1959-049-0 ◽

1959 ◽

Vol 11 ◽

pp. 527-533 ◽

Cited By ~ 10

Author(s):

Fred Brauer

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Uniqueness Theorem ◽

Monotonicity Condition ◽

Uniqueness Of Solutions ◽

Monotonicity Assumption ◽

Logical Independence ◽

Almost All

In the theory of ordinary differential equations, there is a strange relationship between uniqueness of solutions and convergence of the successive approxi mations. There are examples of differential equations with unique solutions for which the successive approximations do not converge (8) and of differential equations with non-unique solutions for which the successive approximations do converge (2). However, in spite of the known logical independence of these two properties, almost all conditions which assure uniqueness also imply the convergence of the successive approximations. For example, the hypotheses of Kamke's general uniqueness theorem (5), have been shown by Coddington and Levinson to suffice for the convergence of successive approximations, after the addition of one simple monotonicity condition (4). There is one counterexample to this “principle,” a generalization of Kamke's result, to which another condition in addition to a monotonicity assumption must be added before convergence of the successive approximations can be proved (2).

Solving Nonlinear Thermal Problems of Friction by Using Method of Lines

Acta Mechanica et Automatica ◽

10.1515/ama-2015-0007 ◽

2015 ◽

Vol 9 (1) ◽

pp. 33-37

Author(s):

Ewa Och

Keyword(s):

Heat Conduction ◽

Differential Equations ◽

Heat Conduction Problem ◽

Method Of Lines ◽

Partial Linearization ◽

The Method Of Lines ◽

Third Stage ◽

Independent Variable

Abstract One-dimensional heat conduction problem of friction for two bodies (half spaces) made of thermosensitive materials was considered. Solution to the nonlinear boundary-value heat conduction problem was obtained in three stages. At the first stage a partial linearization of the problem was performed by using Kirchhoff transform. Next, the obtained boundary-values problem by using the method of lines was brought to a system of nonlinear ordinary differential equations, relatively to Kirchhoff’s function values in the nodes of the grid on the spatial variable, where time is an independent variable. At the third stage, by using the Adams's method from DIFSUB package, a numerical solution was found to the above-mentioned differential equations. A comparative analysis was conducted (Och, 2014) using the results obtained with the proposed method and the method of successive approximations.

Solution of linear systems of differential equations by the use of the method of successive approximations

Annales Polonici Mathematici ◽

10.4064/ap-10-3-309-322 ◽

1961 ◽

Vol 10 (3) ◽

pp. 309-322 ◽

Cited By ~ 3

Author(s):

M. Kwapisz

Keyword(s):

Differential Equations ◽

Linear Systems ◽

Systems Of Differential Equations

On one class of solutions of the quasilinear matrix differential equations

Researches in Mathematics and Mechanics ◽

10.18524/2519-206x.2020.2(36).233806 ◽

2021 ◽

Vol 25 (2(36)) ◽

pp. 95-102

Author(s):

S. A. Shchogolev ◽

V. V. Karapetrov

Keyword(s):

Differential Equations ◽

Quasilinear Equation ◽

Sufficient Conditions ◽

Matrix Equations ◽

Engineering Economics ◽

Matrix Differential Equations ◽

Particular Solution ◽

Matrix Differential

In the mathematical description of various phenomena and processes that arise in mathematical physics, electrical engineering, economics, one has to deal with matrix differential equations. Therefore, these equations are relevant both for mathematicians and for specialists in other areas of natural science. Many studies are devoted to them, in which the solvability of matrix equations in various function spaces, boundary value problems for matrix differential equations, and other problems were investigated. In this article, a quasilinear matrix equation is considered, the coefficients of which can be represented in the form of absolutely and uniformly converging Fourier series with coefficients and frequency slowly varying in a certain sense. The problem is posed of obtaining sufficient conditions for the existence of particular solutions of a similar structure for the equation under consideration. For this purpose, the corresponding linear equation is considered first. It is written down in component-wise form, and, based on the assumptions made, the existence of the only particular solution of the specified structure is proved. Then, using the method of successive approximations and the principle of contracting mappings, the existence of a unique particular solution of the indicated structure for the original quasilinear equation are proved.