scholarly journals Monotone-Iterative Method for the Initial Value Problem with Initial Time Difference for Differential Equations with “Maxima”

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
S. Hristova ◽  
A. Golev
Keyword(s):  
Differential Equations ◽  
Initial Value Problem ◽  
Upper And Lower Solutions ◽  
Successive Approximations ◽  
Initial Time ◽  
Time Interval ◽  
Monotone Iterative Method ◽  
Initial Value ◽  
Monotone Iterative ◽  
The Given

The object of investigation of the paper is a special type of functional differential equations containing the maximum value of the unknown function over a past time interval. An improved algorithm of the monotone-iterative technique is suggested to nonlinear differential equations with “maxima.” The case when upper and lower solutions of the given problem are known at different initial time is studied. Additionally, all initial value problems for successive approximations have both initial time and initial functions different. It allows us to construct sequences of successive approximations as well as sequences of initial functions, which are convergent to the solution and to the initial function of the given initial value problem, respectively. The suggested algorithm is realized as a computer program, and it is applied to several examples, illustrating the advantages of the suggested scheme.

scholarly journals Quasilinearization of the Initial Value Problem for Difference Equations with “Maxima”

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
S. Hristova ◽  
A. Golev ◽  
K. Stefanova
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Initial Value Problem ◽  
Linear Difference Equation ◽  
Successive Approximations ◽  
Time Interval ◽  
Initial Value ◽  
Past Time ◽  
The Given ◽  
Unknown Solution

The object of investigation of the paper is a special type of difference equations containing the maximum value of the unknown function over a past time interval. These equations are adequate models of real processes which present state depends significantly on their maximal value over a past time interval. An algorithm based on the quasilinearization method is suggested to solve approximately the initial value problem for the given difference equation. Every successive approximation of the unknown solution is the unique solution of an appropriately constructed initial value problem for a linear difference equation with “maxima,” and a formula for its explicit form is given. Also, each approximation is a lower/upper solution of the given mixed problem. It is proved the quadratic convergence of the successive approximations. The suggested algorithm is realized as a computer program, and it is applied to an example, illustrating the advantages of the suggested scheme.

A MONOTONE-ITERATIVE METHOD FOR FINDING PERIODIC SOLUTION OF AN IMPULSIVE DIFFERENTIAL EQUATIONS AND APPLICATION

2008 ◽  
Vol 01 (02) ◽  
pp. 247-256
Author(s):  
JIAWEI DOU
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Upper And Lower Solutions ◽  
Impulsive Differential Equations ◽  
Iteration Process ◽  
Impulsive System ◽  
Monotone Iterative Method ◽  
Initial Value ◽  
Monotone Iterative ◽  
Monotone Iterations

In this paper, using the method of upper and lower solutions and its associated monotone iterations, we establish a new monotone-iterative scheme for finding periodic solutions of an impulsive differential equations. This method leads to the existence of maximal and minimal periodic solutions which can be computed from a linear iteration process in the same fashion as for impulsive differential equations initial value problem. This method is constructive and can be used to develop a computational algorithm for numerical solution of the periodic impulsive system. Our existence result improves a result established in [1]. The result is applied to a model of mutualism of Lotka–Volterra type which involves interactions among a mutualist-competitor, a competitor and a mutualist, the existence of positive periodic solutions of the model is obtained.

scholarly journals Monotone-iterative technique of Lakshmikantham for the initial value problem for a differential equation with a step function

2002 ◽  
Vol 30 (1) ◽  
pp. 57-62
Author(s):  
S. G. Hristova ◽  
M. A. Rangelova
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Initial Value Problem ◽  
Special Kind ◽  
Step Function ◽  
Monotone Iterative Technique ◽  
Iterative Technique ◽  
Initial Value ◽  
Monotone Iterative ◽  
The Given

The initial value problem for a special kind of differential equations with a step function is studied. The monotone-iterative technique of Lakshmikantham for approximate finding of the solutions of the given problem is well grounded.

Monotone Iterative Techniques and a Periodic Boundary Value Problem for First Order Differential Equations with a Functional Argument

2000 ◽  
Vol 7 (2) ◽  
pp. 373-378
Author(s):  
Aiqin Qi ◽  
Yansheng Liu
Keyword(s):  
Differential Equations ◽  
Periodic Boundary ◽  
Boundary Value ◽  
Upper And Lower Solutions ◽  
Initial Value ◽  
First Order ◽  
Monotone Iterative ◽  
Periodic Boundary Value Problems ◽  
Functional Argument ◽  
First Order Differential Equations

Abstract This paper is concerned with periodic boundary value problems involving first order differential equations with functional arguments. The main feature of the paper is that the existence of maximal and minimal solutions is obtained by constructing sequences of upper and lower solutions of the initial value problems and not by establishing the comparison principle.

scholarly journals Computer Simulation and Iterative Algorithm for Approximate Solving of Initial Value Problem for Riemann-Liouville Fractional Delay Differential Equations

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 477
Author(s):  
Snezhana Hristova ◽  
Kremena Stefanova ◽  
Angel Golev
Keyword(s):  
Differential Equations ◽  
Initial Value Problem ◽  
Finite Interval ◽  
Initial Value ◽  
Monotone Iterative ◽  
Practical Usefulness ◽  
Fractional Delay ◽  
Fractional Differential ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

The main aim of this paper is to suggest an algorithm for constructing two monotone sequences of mild lower and upper solutions which are convergent to the mild solution of the initial value problem for Riemann-Liouville fractional delay differential equation. The iterative scheme is based on a monotone iterative technique. The suggested scheme is computerized and applied to solve approximately the initial value problem for scalar nonlinear Riemann-Liouville fractional differential equations with a constant delay on a finite interval. The suggested and well-grounded algorithm is applied to a particular problem and the practical usefulness is illustrated.

scholarly journals A monotone iterative method for boundary value problems of parametric differential equations

2001 ◽  
Vol 14 (2) ◽  
pp. 183-187 ◽  
Author(s):  
Xinzhi Liu ◽  
Farzana A. McRae
Keyword(s):  
Differential Equations ◽  
Iterative Method ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Upper And Lower Solutions ◽  
Extremal Solutions ◽  
Monotone Iterative Method ◽  
Monotone Iterative ◽  
Monotone Sequences

This paper studies boundary value problems for parametric differential equations. By using the method of upper and lower solutions, monotone sequences are constructed and proved to converge to the extremal solutions of the boundary value problem.

scholarly journals Initial Value Problem For Nonlinear Fractional Differential Equations With ψ-Caputo Derivative Via Monotone Iterative Technique

Axioms ◽  
2020 ◽  
Vol 9 (2) ◽  
pp. 57 ◽  
Author(s):  
Choukri Derbazi ◽  
Zidane Baitiche ◽  
Mouffak Benchohra ◽  
Alberto Cabada
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Initial Value Problem ◽  
Caputo Derivative ◽  
Monotone Iterative Technique ◽  
Extremal Solutions ◽  
Iterative Technique ◽  
Initial Value ◽  
Monotone Iterative ◽  
Fractional Differential

In this article, we discuss the existence and uniqueness of extremal solutions for nonlinear initial value problems of fractional differential equations involving the ψ -Caputo derivative. Moreover, some uniqueness results are obtained. Our results rely on the standard tools of functional analysis. More precisely we apply the monotone iterative technique combined with the method of upper and lower solutions to establish sufficient conditions for existence as well as the uniqueness of extremal solutions to the initial value problem. An illustrative example is presented to point out the applicability of our main results.

Iterative techniques with computer realization for initial value problems for Riemann–Liouville fractional differential equations

2020 ◽  
Vol 26 (1) ◽  
pp. 21-47 ◽  
Author(s):  
Ravi Agarwal ◽  
A. Golev ◽  
S. Hristova ◽  
D. O’Regan
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Finite Interval ◽  
Lower And Upper Solutions ◽  
Successive Approximations ◽  
Initial Value ◽  
Practical Usefulness ◽  
Computer Environment ◽  
Fractional Differential ◽  
The Given

AbstractThe main aim of this paper is to suggest some algorithms and to use them in an appropriate computer environment to solve approximately the initial value problem for scalar nonlinear Riemann–Liouville fractional differential equations on a finite interval. The iterative schemes are based on appropriately defined lower and upper solutions to the given problem. A number of different cases depending on the type of lower and upper solutions are studied and various schemes for constructing successive approximations are provided. The suggested schemes are applied to some problems and their practical usefulness is illustrated.

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