Research

18 pages, 1418 KiB

Open AccessArticle

Sumudu Decomposition Method for Solving Fuzzy Integro-Differential Equations

by Shin Min Kang, Zain Iqbal, Mustafa Habib and Waqas Nazeer

Axioms 2019, 8(2), 74; https://doi.org/10.3390/axioms8020074 - 20 Jun 2019

Cited by 10 | Viewed by 2982

Different results regarding different integro-differentials are usually not properly generalized, as they often do not satisfy some of the constraints. The field of fuzzy integro-differentials is very rich these days because of their different applications and functions in different physical phenomena. Solutions of [...] Read more.

Different results regarding different integro-differentials are usually not properly generalized, as they often do not satisfy some of the constraints. The field of fuzzy integro-differentials is very rich these days because of their different applications and functions in different physical phenomena. Solutions of linear fuzzy Volterra integro-differential equations (FVIDEs) are more generalized and have better applications. In this report, the Sumudu decomposition method (SDM) was used to find the solution to some linear and nonlinear fuzzy integro-differential equations (FIDEs). Some examples are given to show the validity of the presented method. Full article

(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)

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11 pages, 252 KiB

Open AccessArticle

On the Polynomial Solution of Divided-Difference Equations of the Hypergeometric Type on Nonuniform Lattices

by Mama Foupouagnigni and Salifou Mboutngam

Axioms 2019, 8(2), 47; https://doi.org/10.3390/axioms8020047 - 21 Apr 2019

Cited by 3 | Viewed by 2285

Abstract

In this paper, we provide a formal proof of the existence of a polynomial solution of fixed degree for a second-order divided-difference equation of the hypergeometric type on non-uniform lattices, generalizing therefore previous work proving existence of the polynomial solution for second-order differential, [...] Read more.

In this paper, we provide a formal proof of the existence of a polynomial solution of fixed degree for a second-order divided-difference equation of the hypergeometric type on non-uniform lattices, generalizing therefore previous work proving existence of the polynomial solution for second-order differential, difference or q-difference equation of hypergeometric type. This is achieved by studying the properties of the mean operator and the divided-difference operator as well as by defining explicitly, the right and the “left” inverse for the second operator. The method constructed to provide this formal proof is likely to play an important role in the characterization of orthogonal polynomials on non-uniform lattices and might also be used to provide hypergeometric representation (when it does exist) of the second solution—non polynomial solution—of a second-order divided-difference equation of hypergeometric type. Full article

(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)

18 pages, 297 KiB

Open AccessArticle

Bäcklund Transformations for Nonlinear Differential Equations and Systems

by Tatyana V. Redkina, Robert G. Zakinyan, Arthur R. Zakinyan, Olesya B. Surneva and Olga S. Yanovskaya

Axioms 2019, 8(2), 45; https://doi.org/10.3390/axioms8020045 - 11 Apr 2019

Cited by 3 | Viewed by 2386

Abstract

In this work, new Bäcklund transformations (BTs) for generalized Liouville equations were obtained. Special cases of Liouville equations with exponential nonlinearity that have a multiplier that depends on the independent variables and first-order derivatives from the function were considered. Two- and three-dimensional cases [...] Read more.

In this work, new Bäcklund transformations (BTs) for generalized Liouville equations were obtained. Special cases of Liouville equations with exponential nonlinearity that have a multiplier that depends on the independent variables and first-order derivatives from the function were considered. Two- and three-dimensional cases were considered. The BTs construction is based on the method proposed by Clairin. The solutions of the considered equations have been found using the BTs, with a unified algorithm. In addition, the work develops the Clairin’s method for the system of two third-order equations related to the integrable perturbation and complexification of the Korteweg-de Vries (KdV) equation. Among the constructed BTs an analog of the Miura transformations was found. The Miura transformations transfer the initial system to that of perturbed modified KdV (mKdV) equations. It could be shown on this way that, considering the system as a link between the real and imaginary parts of a complex function, it is possible to go to the complexified KdV (cKdV) and here the analog of the Miura transformations transforms it into the complexification of the mKdV. Full article

(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)

51 pages, 471 KiB

Open AccessArticle

The 3D Navier–Stokes Equations: Invariants, Local and Global Solutions

by Vladimir I. Semenov

Axioms 2019, 8(2), 41; https://doi.org/10.3390/axioms8020041 - 07 Apr 2019

Cited by 2 | Viewed by 2140

Abstract

In this article, I consider local solutions of the 3D Navier–Stokes equations and its properties such as an existence of global and smooth solution, uniform boundedness. The basic role is assigned to a special invariant class of solenoidal vector fields and three parameters [...] Read more.

In this article, I consider local solutions of the 3D Navier–Stokes equations and its properties such as an existence of global and smooth solution, uniform boundedness. The basic role is assigned to a special invariant class of solenoidal vector fields and three parameters that are invariant with respect to the scaling procedure. Since in spaces of even dimensions the scaling procedure is a conformal mapping on the Heisenberg group, then an application of invariant parameters can be considered as the application of conformal invariants. It gives the possibility to prove the sufficient and necessary conditions for existence of a global regular solution. This is the main result and one among some new statements. With some compliments, the rest improves well-known classical results. Full article

(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)

11 pages, 275 KiB

Open AccessArticle

First Order Coupled Systems With Functional and Periodic Boundary Conditions: Existence Results and Application to an SIRS Model

by João Fialho and Feliz Minhós

Axioms 2019, 8(1), 23; https://doi.org/10.3390/axioms8010023 - 16 Feb 2019

Cited by 3 | Viewed by 3001

Abstract

The results presented in this paper deal with the existence of solutions of a first order fully coupled system of three equations, and they are split in two parts: 1. Case with coupled functional boundary conditions, and 2. Case with periodic boundary conditions. [...] Read more.

The results presented in this paper deal with the existence of solutions of a first order fully coupled system of three equations, and they are split in two parts: 1. Case with coupled functional boundary conditions, and 2. Case with periodic boundary conditions. Functional boundary conditions, which are becoming increasingly popular in the literature, as they generalize most of the classical cases and in addition can be used to tackle global conditions, such as maximum or minimum conditions. The arguments used are based on the Arzèla Ascoli theorem and Schauder’s fixed point theorem. The existence results are directly applied to an epidemic SIRS (Susceptible-Infectious-Recovered-Susceptible) model, with global boundary conditions. Full article

(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)

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14 pages, 272 KiB

Open AccessArticle

Heteroclinic Solutions for Classical and Singular ϕ-Laplacian Non-Autonomous Differential Equations

by Feliz Minhós

Axioms 2019, 8(1), 22; https://doi.org/10.3390/axioms8010022 - 15 Feb 2019

Cited by 2 | Viewed by 2116

Abstract

In this paper, we consider the second order discontinuous differential equation in the real line, [...] Read more.

In this paper, we consider the second order discontinuous differential equation in the real line,

{(a (t, u) ϕ (u^{'}))}^{'} = f (t, u, u^{'}), a . e . t \in R, u (- \infty) = ν^{-}, u (+ \infty) = ν^{+},

with

ϕ

an increasing homeomorphism such that

ϕ (0) = 0

and

ϕ (R) = R

,

a \in C (R^{2}, R)

with

a (t, x) > 0

for

(t, x) \in R^{2}

,

f : R^{3} \to R

a

L^{1}

-Carathéodory function and

ν^{-}, ν^{+} \in R

such that

ν^{-} < ν^{+}

. The existence and localization of heteroclinic connections is obtained assuming a Nagumo-type condition on the real line and without asymptotic conditions on the nonlinearities

ϕ

and

f

. To the best of our knowledge, this result is even new when

ϕ (y) = y

, that is for equation

{(a (t, u (t)) u^{'} (t))}^{'} = f (t, u (t), u^{'} (t)), a . e . t \in R

. Moreover, these results can be applied to classical and singular

ϕ

-Laplacian equations and to the mean curvature operator. Full article

(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)

22 pages, 317 KiB

Open AccessArticle

Solution Estimates for the Discrete Lyapunov Equation in a Hilbert Space and Applications to Difference Equations

by Michael Gil’

Axioms 2019, 8(1), 20; https://doi.org/10.3390/axioms8010020 - 06 Feb 2019

Cited by 6 | Viewed by 2647

Abstract

The paper is devoted to the discrete Lyapunov equation

X - A^{*} X A = C

, where A and C are given operators in a Hilbert space

H

and X should be found. We derive norm estimates for solutions of that [...] Read more.

The paper is devoted to the discrete Lyapunov equation

X - A^{*} X A = C

, where A and C are given operators in a Hilbert space

H

and X should be found. We derive norm estimates for solutions of that equation in the case of unstable operator A, as well as refine the previously-published estimates for the equation with a stable operator. By the point estimates, we establish explicit conditions, under which a linear nonautonomous difference equation in

H

is dichotomic. In addition, we suggest a stability test for a class of nonlinear nonautonomous difference equations in

H

. Our results are based on the norm estimates for powers and resolvents of non-self-adjoint operators. Full article

(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)

10 pages, 327 KiB

Open AccessEditor’s ChoiceArticle

Note on Limit-Periodic Solutions of the Difference Equation x_{t + 1} − [h(x_t) + λ]x_t = r_t, λ > 1

by Jan Andres and Denis Pennequin

Axioms 2019, 8(1), 19; https://doi.org/10.3390/axioms8010019 - 05 Feb 2019

Cited by 2 | Viewed by 2759

Abstract

As a nontrivial application of the abstract theorem developed in our recent paper titled “Limit-periodic solutions of difference and differential systems without global Lipschitzianity restricitions”, the existence of limit-periodic solutions of the difference equation from the title is proved, both in the scalar [...] Read more.

As a nontrivial application of the abstract theorem developed in our recent paper titled “Limit-periodic solutions of difference and differential systems without global Lipschitzianity restricitions”, the existence of limit-periodic solutions of the difference equation from the title is proved, both in the scalar as well as vector cases. The nonlinearity h is not necessarily globally Lipschitzian. Several simple illustrative examples are supplied. Full article

(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)

17 pages, 2802 KiB

Open AccessArticle

Lipschitz Stability for Non-Instantaneous Impulsive Caputo Fractional Differential Equations with State Dependent Delays

by Ravi Agarwal, Snezhana Hristova and Donal O’Regan

Axioms 2019, 8(1), 4; https://doi.org/10.3390/axioms8010004 - 29 Dec 2018

Cited by 8 | Viewed by 2620

Abstract

In this paper, we study Lipschitz stability of Caputo fractional differential equations with non-instantaneous impulses and state dependent delays. The study is based on Lyapunov functions and the Razumikhin technique. Our equations in particular include constant delays, time variable delay, distributed delay, etc. [...] Read more.

In this paper, we study Lipschitz stability of Caputo fractional differential equations with non-instantaneous impulses and state dependent delays. The study is based on Lyapunov functions and the Razumikhin technique. Our equations in particular include constant delays, time variable delay, distributed delay, etc. We consider the case of impulses that start abruptly at some points and their actions continue on given finite intervals. The study of Lipschitz stability by Lyapunov functions requires appropriate derivatives among fractional differential equations. A brief overview of different types of derivative known in the literature is given. Some sufficient conditions for uniform Lipschitz stability and uniform global Lipschitz stability are obtained by an application of several types of derivatives of Lyapunov functions. Examples are given to illustrate the results. Full article

(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)

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13 pages, 782 KiB

Open AccessArticle

A New Efficient Method for the Numerical Solution of Linear Time-Dependent Partial Differential Equations

by Mina Torabi and Mohammad-Mehdi Hosseini

Axioms 2018, 7(4), 70; https://doi.org/10.3390/axioms7040070 - 01 Oct 2018

Cited by 4 | Viewed by 2716

Abstract

This paper presents a new efficient method for the numerical solution of a linear time-dependent partial differential equation. The proposed technique includes the collocation method with Legendre wavelets for spatial discretization and the three-step Taylor method for time discretization. This procedure is third-order [...] Read more.

This paper presents a new efficient method for the numerical solution of a linear time-dependent partial differential equation. The proposed technique includes the collocation method with Legendre wavelets for spatial discretization and the three-step Taylor method for time discretization. This procedure is third-order accurate in time. A comparative study between the proposed method and the one-step wavelet collocation method is provided. In order to verify the stability of these methods, asymptotic stability analysis is employed. Numerical illustrations are investigated to show the reliability and efficiency of the proposed method. An important property of the presented method is that unlike the one-step wavelet collocation method, it is not necessary to choose a small time step to achieve stability. Full article

(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)

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