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Quasilinearization method for finite systems of nonlinear RL fractional differential equations

Denton Zachary, Ramirez Juan Diego

Opuscula Mathematica

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2020

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Vol. 40, no. 6

667–-683

EN

In this paper the quasilinearization method is extended to finite systems of Riemann-Liouville fractional differential equations of order 0 < q < 1. Existence and comparison results of the linear Riemann-Liouville fractional differential systems are recalled and modified where necessary. Using upper and lower solutions, sequences are constructed that are monotonic such that the weighted sequences converge uniformly and quadratically to the unique solution of the system. A numerical example illustrating the main result is given.

2

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

Agarwal Ravi, Golev A., Hristova S., O’Regan D.

Journal of Applied Analysis

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2020

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Vol. 26, nr 1

21--47

EN

The 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.

3

Monotone method for Riemann-Liouville multi-order fractional differential systems

Denton Z.

Opuscula Mathematica

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2016

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Vol. 36, no. 2

189--206

EN

In this paper we develop the monotone method for nonlinear multi-order N-systems of Riemann-Liouville fractional differential equations. That is, a hybrid system of nonlinear equations of orders qi where 0 < qi < 1. In the development of this method we recall any needed existence results along with any necessary changes. Through the method's development we construct a generalized multi-order Mittag-Leffler function that fulfills exponential-like properties for multi-order systems. Further we prove a comparison result paramount for the discussion of fractional multi-order inequalities that utilizes lower and upper solutions of the system. The monotone method is then developed via the construction of sequences of linear systems based on the upper and lower solutions, and are used to approximate the solution of the original nonlinear multi-order system.

4

On coupled first and second order boundary value problems in ordered Banach spaces

Herzog G., Lemmert R.

Demonstratio Mathematica

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2010

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Vol. 43, nr 3

561-574

EN

By means of a result on coupled first and second order differential inequalities and an intermediate value theorem in ordered Banach spaces, we obtain the existence of extremal solutions of boundary value problems of the form u܉ = f(t, u1, u2), u𔈀 + g(t, u1, u2) = 0, u1(a) = xa, u2(a) = ya, u2(b) = yb, between lower and upper solutions.

5

Existence theorems for a fourth order boundary value problem

El-Haffaf A.

Bulletin of the Polish Academy of Sciences. Mathematics

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2009

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Vol. 57, no 2

135-148

EN

This paper treats the question of the existence of solutions of a fourth order boundary value problem having the following form: x(4)(t) + ƒ(t,x(t),x''(t)) = 0, 0 < t < 1, x(0) = x'(0) = 0, x''(1) = 0, x(3)(1) = 0. Boundary value problems of very similar type are also considered. It is assumed that ƒ is a function from the space C([0, 1] x R2,R). The main tool used in the proof is the Leray-Schauder nonlinear alternative.