scholarly journals New results and applications on the existence results for nonlinear coupled systems

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Imran Talib ◽  
Thabet Abdeljawad ◽  
Manar A. Alqudah ◽  
Cemil Tunc ◽  
Rabia Ameen
Keyword(s):  
Fixed Point Theory ◽  
Coupled System ◽  
Point Theory ◽  
Coupled Systems ◽  
Lower And Upper Solutions ◽  
Nonlinear Boundary Value Problems ◽  
Fully Nonlinear ◽  
Nonlinear Coupled System ◽  
A Priori Bound ◽  
Special Cases

AbstractIn this manuscript, we study a certain classical second-order fully nonlinear coupled system with generalized nonlinear coupled boundary conditions satisfying the monotone assumptions. Our new results unify the existence criteria of certain linear and nonlinear boundary value problems (BVPs) that have been previously studied on a case-by-case basis; for example, Dirichlet and Neumann are special cases. The common feature is that the solution of each BVPs lies in a sector defined by well-ordered coupled lower and upper solutions. The tools we use are the coupled lower and upper solutions approach along with some results of fixed point theory. By means of the coupled lower and upper solutions approach, the considered BVPs are logically modified to new problems, known as modified BVPs. The solution of the modified BVPs leads to the solution of the original BVPs. In our case, we only require the Nagumo condition to get a priori bound on the derivatives of the solution function. Further, we extend the results presented in (Franco et al. in Extr. Math. 18(2):153–160, 2003; Franco et al. in Appl. Math. Comput. 153:793–802, 2004; Franco and O’Regan in Arch. Inequal. Appl. 1:423–430, 2003; Asif et al. in Bound. Value Probl. 2015:134, 2015). Finally, as an application, we consider the fully nonlinear coupled mass-spring model.

Hermite collocation method for obtaining the chaotic behavior of a nonlinear coupled system of FDEs

2020 ◽  
Vol 31 (07) ◽  
pp. 2050093
Author(s):  
M. M. Khader ◽  
Mohammed M. Babatin
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Chaotic Behavior ◽  
Standard Form ◽  
Coupled System ◽  
Coupled Systems ◽  
Electrical Network ◽  
Algebraic Equations ◽  
Home Heating ◽  
Nonlinear Coupled System

This paper is devoted to introduce an efficient solver using the Hermite collocation technique (HCT), of the coupled system of fractional differential equations (FDEs). The given systems are of basic importance in modeling various phenomena like Cascades and Compartment Analysis, Pond Pollution, Home Heating, Chemostats, and Microorganism Culturing, Nutrient Flow in an Aquarium, Biomass Transfer, Forecasting Prices, Electrical Network, Earthquake Effects on Buildings. The proposed method reduces the system of FDEs to a system of algebraic equations in the coefficients of the expansion using the Hermite polynomials. The introduced method is computer oriented and provides highly accurate solution. To demonstrate the efficiency of the method, two examples are solved and the results are displayed graphically. Finally, we convert the presented coupled systems from the case of its standard form to a first-order ordinary differential equations to compare the obtained numerical solutions with those solutions using the fourth-order Runge–Kutta method (RK4).

scholarly journals Homotopy Series Solutions to Time-Space Fractional Coupled Systems

2017 ◽  
Vol 2017 ◽  
pp. 1-19 ◽  
Author(s):  
Jin Zhang ◽  
Ming Cai ◽  
Bochao Chen ◽  
Hui Wei
Keyword(s):  
Water System ◽  
Coupled System ◽  
Coupled Systems ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Time Space ◽  
Nonlinear Coupled System ◽  
Sumudu Transform ◽  
Shallow Water System ◽  
Burgers System

We apply the homotopy perturbation Sumudu transform method (HPSTM) to the time-space fractional coupled systems in the sense of Riemann-Liouville fractional integral and Caputo derivative. The HPSTM is a combination of Sumudu transform and homotopy perturbation method, which can be easily handled with nonlinear coupled system. We apply the method to the coupled Burgers system, the coupled KdV system, the generalized Hirota-Satsuma coupled KdV system, the coupled WBK system, and the coupled shallow water system. The simplicity and validity of the method can be shown by the applications and the numerical results.

scholarly journals Some Logical Metatheorems with Applications in Functional Analysis

2003 ◽  
Vol 10 (21) ◽  
Author(s):  
Ulrich Kohlenbach
Keyword(s):  
Functional Analysis ◽  
Fixed Point ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Inner Product ◽  
Original Proof ◽  
Uniform Bounds ◽  
Existence Proofs ◽  
Special Cases

In previous papers we have developed proof-theoretic techniques for extracting effective uniform bounds from large classes of ineffective existence proofs in functional analysis. `Uniform' here means independence from parameters in compact spaces. A recent case study in fixed point theory systematically yielded uniformity even w.r.t. parameters in metrically bounded (but noncompact) subsets which had been known before only in special cases. In the present paper we prove general logical metatheorems which cover these applications to fixed point theory as special cases but are not restricted to this area at all. Our theorems guarantee under general logical conditions such strong uniform versions of non-uniform existence statements. Moreover, they provide algorithms for actually extracting effective uniform bounds and transforming the original proof into one for the stronger uniformity result. Our metatheorems deal with general classes of spaces like metric spaces, hyperbolic spaces, normed linear spaces, uniformly convex spaces as well as inner product spaces.

scholarly journals Viscosity solutions of fully nonlinear functional parabolic PDE

2005 ◽  
Vol 2005 (22) ◽  
pp. 3539-3550
Author(s):  
Liu Wei-an ◽  
Lu Gang
Keyword(s):  
Optimal Control ◽  
Viscosity Solutions ◽  
Parabolic Equations ◽  
Optimal Control Theory ◽  
Existence And Uniqueness ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Fully Nonlinear ◽  
Nonlinear Functional ◽  
Parabolic Pde

By the technique of coupled solutions, the notion of viscosity solutions is extended to fully nonlinear retarded parabolic equations. Such equations involve many models arising from optimal control theory, economy and finance, biology, and so forth. The comparison principle is shown. Then the existence and uniqueness are established by the fixed point theory.

scholarly journals Fractional Differential Equation Involving Mixed Nonlinearities with Nonlocal Multi-Point and Riemann-Stieltjes Integral-Multi-Strip Conditions

2019 ◽  
Vol 3 (2) ◽  
pp. 34 ◽  
Author(s):  
Ahmad ◽  
Alsaedi ◽  
Salem ◽  
Ntouyas
Keyword(s):  
Fixed Point Theory ◽  
Point Theory ◽  
Stieltjes Integral ◽  
Existence And Uniqueness Results ◽  
New Class ◽  
Uniqueness Results ◽  
Special Cases ◽  
Mixed Nonlinearities ◽  
Fractional Differential ◽  
The Given

In this paper, we investigate a new class of boundary value problems involving fractionaldifferential equations with mixed nonlinearities, and nonlocal multi-point and Riemann–Stieltjesintegral-multi-strip boundary conditions. Based on the standard tools of the fixed point theory,we obtain some existence and uniqueness results for the problem at hand, which are well illustratedwith the aid of examples. Our results are not only in the given configuration but also yield severalnew results as special cases. Some variants of the given problem are also discussed.

scholarly journals Functional Coupled Systems with Generalized Impulsive Conditions and Application to a SIRS-Type Model

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Feliz Minhós ◽  
Rui Carapinha
Keyword(s):  
Fixed Point Theory ◽  
Impulsive System ◽  
Point Theory ◽  
Coupled Systems ◽  
Impulsive Effects ◽  
Solution Technique ◽  
Type Model ◽  
Sirs Model ◽  
Lower And Upper Solution ◽  
Functional Boundary

In this paper, we consider a first-order coupled impulsive system of equations with functional boundary conditions, subject to the generalized impulsive effects. It is pointed out that this problem generalizes the classical boundary assumptions, allowing two-point or multipoint conditions, nonlocal and integrodifferential ones, or global arguments, as maxima or minima, among others. Our method is based on lower and upper solution technique together with the fixed point theory. The main theorem is applied to a SIRS model where to the best of our knowledge, for the first time, it includes impulsive effects combined with global, local, and the asymptotic behavior of the unknown functions.

scholarly journals A Finite Element Approach to the Analysis of Rotating Bladed-Disk Assemblies Coupled With Flexible Shaft

1994 ◽  
Author(s):  
Wen Zhang ◽  
Wenliang Wang ◽  
Hao Wang ◽  
Jiong Tang
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Coupled System ◽  
Equation Of Motion ◽  
Coupled Systems ◽  
The Finite Element Method ◽  
Bladed Disk ◽  
Modal Synthesis ◽  
Element Approach ◽  
Final Equation

A method for dynamic analysis of flexible bladed-disk/shaft coupled systems is presented in this paper. Being independant substructures first, the rigid-disk/shaft and each of the bladed-disk assemblies are analyzed separately in a centrifugal force field by means of the finite element method. Then through a modal synthesis approach the equation of motion for the integral system is derived. In the vibration analysis of the rotating bladed-disk substructure, the geometrically nonlinear deformation is taken into account and the rotationally periodic symmetry is utilized to condense the degrees of freedom into one sector. The final equation of motion for the coupled system involves the degrees of freedom of the shaft and those of only one sector of each of the bladed-disks, thereby reducing the computer storage. Some computational and experimental results are given.

scholarly journals Existence and uniqueness of positive solutions for nonlinear fractional mixed problems

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Alberto Cabada ◽  
Om Kalthoum Wanassi
Keyword(s):  
Existence And Uniqueness ◽  
Fixed Point Theory ◽  
Mixed Boundary ◽  
Point Theory ◽  
Uniqueness Of Solutions ◽  
Nonlinear Part ◽  
Banach Contraction ◽  
Fractional Differential ◽  
Exhaustive Study ◽  
The Banach Contraction Principle

Abstract This paper is devoted to study the existence and uniqueness of solutions of a one parameter family of nonlinear Riemann–Liouville fractional differential equations with mixed boundary value conditions. An exhaustive study of the sign of the related Green’s function is carried out. Under suitable assumptions on the asymptotic behavior of the nonlinear part of the equation at zero and at infinity, and by application of the fixed point theory of compact operators defined in suitable cones, it is proved that there exists at least one solution of the considered problem. Moreover, the method of lower and upper solutions is developed and the existence of solutions is deduced by a combination of both techniques. In particular cases, the Banach contraction principle is used to ensure the uniqueness of solutions.

scholarly journals Quasi-stability and continuity of attractors for nonlinear system of wave equations

2021 ◽  
Vol 8 (1) ◽  
pp. 27-45
Author(s):  
M. M. Freitas ◽  
M. J. Dos Santos ◽  
A. J. A. Ramos ◽  
M. S. Vinhote ◽  
M. L. Santos
Keyword(s):  
Wave Equations ◽  
Coupled System ◽  
Global Attractors ◽  
Time Behavior ◽  
Exponential Attractors ◽  
Small Perturbations ◽  
Recent Approach ◽  
Nonlinear Coupled System ◽  
Long Time ◽  
External Forces

Abstract In this paper, we study the long-time behavior of a nonlinear coupled system of wave equations with damping terms and subjected to small perturbations of autonomous external forces. Using the recent approach by Chueshov and Lasiecka in [21], we prove that this dynamical system is quasi-stable by establishing a quasistability estimate, as consequence, the existence of global and exponential attractors is proved. Finally, we investigate the upper and lower semicontinuity of global attractors under autonomous perturbations.

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