Nonlinear Implicit Differential Equations Research Articles

Existence of Solutions for a Coupled System of Nonlinear Implicit Differential Equations Involving $$\varrho $$-Fractional Derivative with Anti Periodic Boundary Conditions

Existence of Solutions for a Coupled System of Nonlinear Implicit Differential Equations Involving $$\varrho $$-Fractional Derivative with Anti Periodic Boundary Conditions

Stability of Nonlinear Implicit Differential Equations with Caputo–Katugampola Fractional Derivative

The purpose of this paper is to study nonlinear implicit differential equations with the Caputo–Katugampola fractional derivative. By using Gronwall inequality and Banach fixed-point theorem, the existence of the solution of the implicit equation is proved, and the relevant conclusions about the stability of Ulam–Hyers are obtained. Finally, the correctness of the conclusions is verified by an example.

Ulam stability for nonlinear implicit differential equations with Hilfer-Katugampola fractional derivative and impulses

<abstract><p>In this paper, we investigate the existence, uniqueness and stability results for a class of nonlinear impulsive Hilfer-Katugampola problems. Our reasoning is founded on the Banach contraction principle and Krasnoselskii's fixed point theorem. In addition, an example is provided to demonstrate the effectiveness of the main results.</p></abstract>

On fractional boundary value problems involving fractional derivatives with Mittag-Leffler kernel and nonlinear integral conditions

In this paper, we consider two classes of boundary value problems for nonlinear implicit differential equations with nonlinear integral conditions involving Atangana–Baleanu–Caputo fractional derivatives of orders 0<vartheta leq 1 and 1<vartheta leq 2. We structure the equivalent fractional integral equations of the proposed problems. Further, the existence and uniqueness theorems are proved with the aid of fixed point theorems of Krasnoselskii and Banach. Lastly, the paper includes pertinent examples to justify the validity of the results.

THEORY OF FRACTIONAL IMPLICIT DIFFERENTIAL EQUATIONS WITH COMPLEX ORDER

In this paper, we consider boundary value problems for the following nonlinear implicit differential equations with complex order D +x(t) = f t,x(t),D +x(t) , ? = m+i?, t ? J := [0,T], ax(0)+bx(T) = c,where D + is the Caputo fractional derivative of order ? ? C. Let ? ? R , 0 &amp;lt; ? &amp;lt; 1, m ? (0,1], and f : J ×R ? R is given continuous function. Here a,b,c are real constants with a+b = 0. We derive the existence and stability of solution for a class of boundary value problem(BVP) for nonlinear fractional implicit differential equations(FIDEs) with complex order. The results are based upon the Banach contraction principle and Schaefer’s fixed point theorem.

Existence and Ulam stability for nonlinear implicit differential equations with Riemann-Liouville fractional derivative

Abstract In this paper, we establish the existence and uniqueness of solutions for a class of initial value problem for nonlinear implicit fractional differential equations with Riemann-Liouville fractional derivative, also, the stability of this class of problem. The arguments are based upon the Banach contraction principle and Schaefer’s fixed point theorem. An example is included to show the applicability of our results.

A Simple Method for Impasse Points Detection in Nonlinear Electrical Circuits

In a nonlinear system, impasse points are singularities beyond which solutions are not continuable. In this article, we study two families of nonlinear electrical circuits, which can be represented by nonlinear Implicit Differential Equations. We set conditions that ensure the existence of impasse points in both families of circuits. In the literature, there exist general results to analyse the presence of such singularities in given differential equations of this type. However, the method proposed in this work allows detecting their existence in these electrical topologies in an extremely straightforward way, as illustrated by the examples of application.

Existence of Solutions for Nonlinear Implicit Differential Equations: An Equilibrium Problem Approach

ABSTRACTIn this article, we study the existence of solutions for nonlinear implicit differential equations associated to a time-dependent pseudomonotone (respectively, quasimonotone) operator by using a new approach based on the theory of equilibrium \\hboxproblems. More precisely, we first establish some existence results of solutions for mixed equilibrium problems where the \\hboxbifunctions are, respectively, maximal monotone and \\hboxpseudomonotone (quasimonotone) in the topological sense. Then, we write the nonlinear implicit differential equations in the form of a mixed equilibrium problem. By using the existence results for mixed equilibrium problems, we establish the existence of solutions of nonlinear implicit differential equations. This new approach provides some new and nice results which improve and unify most of the recent results obtained in this direction.

On controlling objects whose motion is defined by implicit nonlinear differential equations

We prove a new result on Lipshitz disturbances of vector covering maps in metric spaces. With methods of covering maps theory, we study controllable systems defined by differential equations that are not resolved with respect to the derivative of the unknown function.

63176 A Graph-Theoretic Approach to Large Scale Multibody Systems(Modeling, Formalisms, and DAE Solution Method)

Mathematical models of complicated mechanical systems such as multibody systems with kinematic constraints, whether holonomic or nonholonomic, are generally represented by implicit nonlinear Differential-Algebraic Equations (DAEs). For the numerical integration of the constrained multibody dynamics represented by the DAEs, we eventually need to calculate inversion of a large scale sparse Jacobian matrix during Newton iteration at each time step. This directly causes in taking much CPU time, since the Jacobian matrix of such mathematical models has generally the characteristic of random sparseness and hence it is very difficult to solve the matrix inversion fast unless employing some efficient sparse matrix technique based on topological structure of the DAEs. In this paper, we propose a graph theoretic approach to symbolic generation of sparse matrix inversion for large scale multibody systems, by which one can "symbolically" calculate matrix inversions of random sparse Jacobian matrices associated to nonlinear implicit differential equations for the sake of fast numerical integrations. It is noteworthy that one can easily calculate the sparse matrix inversion without numerical Gaussian elimination at each time step. To do this, we first show how kinematical and dynamical relations can be effectively set up by introducing connection matrices by mechanical analogy with KCL and KVL constraints in circuit theory. Second, we show how to assign input-output relations among variables in all the kinematical and dynamical relations that appear in the mathematical models of DAEs by introducing bipartite graphs. Then, we demonstrate how one can choose pivots of the Jacobian matrix associated to the kinematical and dynamical relations by using input-output relations in the context of bipartite graphs. Furthermore, we explain solvability for the sparse Jacobian matrix inversion associated to the DAEs by using the bipartite graph. Finally, we propose symbolic generation of the sparse matrix inversion for the Jacobian matrix, which is to be explicitly done by symbolic manipulation and we demonstrate the validity of the proposed approach in numerical efficiency by an example of the stanford manipulator.

On discrete analogues of nonlinear implicit differential equations

This paper deals with some classes of nonlinear implicit difference equations obtained via discretization of nonlinear differential-algebraic or partial differential-algebraic equations. The unique solvability of discretized problems is proved and the compatibility between index notions for nonlinear differential-algebraic equations and nonlinear implicit difference equations is studied.

Free vibrations of cracked prestressed concrete beams

The paper deals with the effects of cracks and the nonlinear material behavior of concrete and prestress reinforcement on the natural frequencies of cracked bonded prestressed beams. The natural frequencies are determined at different levels of crack damage, which are a result of an external static load. The governing dynamic equations of the prestressed beam and their associated boundary and continuity conditions are derived using the variational principle of virtual work. The nonlinear material behavior of the concrete and prestress reinforcement, and the effects of cracking are modeled using the nonlinear constitutive relations of the various materials, which are implemented into an incremental formulation. The formulation yields a set of implicit nonlinear differential equations, which are solved using a linearization procedure and the Multiple Shooting method. Numerical examples are studied to investigate the effects of cracks and the nonlinear behavior of the concrete and prestress reinforcement on the natural frequencies of cracked prestressed beams. It appears that large cracking damage yields drastic reduction in the natural frequencies.

SLIDING FRICTION-INDUCED NON-LINEARITY AND PARAMETRIC EFFECTS IN GEAR DYNAMICS

The mechanism of gear meshing entails a large amount of sliding between the mating teeth in contact. Friction forces act orthogonal to the line of action, and the resulting dynamic force and moment are governed by a number of parameters, such as the relative surface speed, instantaneous load and spatial location of the point of contact. Sliding resistance is inherently non-linear in nature, and due to the additional presence of periodic meshing properties like stiffness and viscous damping, dynamic interactions result between friction and system parameters. This combination leads to a non-linear time-varying (NLTV) system comprising implicit non-linear differential equations. Nevertheless, most researchers have treated friction as a linear time-varying (LTV) phenomenon. In this article, both of these models are critically analyzed. For the LTV system, the harmonic balance formulation is developed to predict the dynamic behavior and sub-harmonic instabilities in the system. The NLTV analysis leads to very complex and intractable equations, and hence numerical methods are applied. Using this analysis, the physical phenomena associated with the two models are compared and the essential differences in the system behavior are examined. Finally, the dynamic effects of friction-induced non-linearity are investigated and the critical parameters are identified.

Lumped Kinetics for Solid-Catalyzed Wet Oxidation: A Versatile Model

Kinetic modeling of solid-catalyzed deep oxidation of pollutants in water is crucial to the design and scale-up of wastewater wet oxidation treatment. Due to their simplicity, the overall kinetics using power-law rates are often unable to capture the important features in such oxidation systems. However, detailed mechanistic approaches aimed at establishing complex reaction networks where several species and intermediates need to be identified become quickly cumbersome and costly. Lumped kinetic approaches offer a balanced trade-off between sophistication and simplicity. A versatile “three-lump” triangular kinetic model was proposed for the description of solid-catalyzed wet oxidation of various pollutants in wastewater effluents. The model is an offshoot of the well-known “generalized lumped kinetic model” introduced for homogeneous wet oxidation. This model was extended, using the Langmuir–Hinshelwood–Hougen–Watson framework, to describe the evolution of heterogeneous catalytic wet oxidation reactions. The model, in the form of a set of three implicit nonlinear differential equations, was validated using literature data obtained under a variety of experimental conditions, such as subcritical or supercritical water conditions, batch and continuous reactors, a multitude of organic loads in the form of carbon-, nitrogen-, and oxygen-bearing compounds, and using different kinetic variables such as TOC and COD. In all cases, this strategy led to calculated parameters that met the thermodynamic, kinetic, and statistical criteria. The uncertainty and confidence joint regions were estimated using bootstrap “Monte Carlo” techniques.

NUMERICAL SOLUTIONS OF SECOND ORDER IMPLICIT NON-LINEAR ORDINARY DIFFERENTIAL EQUATIONS

The existing literature usually assumes that second order ordinary differential equations can be put in first order form, and this assumption is the starting point of most treatments of ordinary differential equations. This paper examines numerical schemes for solving second order implicitnon-linear differential equations. Based on the literature review, three specific methods have been selected here. Two of them are based on series expansions: Picard iteration using Chebyshev series and Incremental Harmonic Balance (IHB), in which the non-linear differential equation is transformed into a set of algebraic ones that are solved iteratively. The third method is based on a fourth order (Houbolt's scheme) and an eighth order backward finite difference method (FDM). Each method is presented first, and then applied to specific examples. It is shown: (1) that the Picard method is not valid for solving implicit equations containing large non-linear inertial terms; (2) that the IHB method yields accurate periodic solutions, together with the frequency of oscillation, and the dynamical stability of the system may be assessed very easily; and (3) that both Houbolt's and the eighth order scheme can be used to compute time histories of initial value problems, if the time step is properly chosen. Finally, it is also illustrated how the combination of IHB and FDM can be a powerful tool for the analysis of non-linear vibration problems defined by implicit differential equations (including also explicit ones), since bifurcation diagrams of stable and unstable periodic solutions can be computed easily with IHB, while periodic and non-periodic stable oscillations may be obtained with FDM.

Index and Decomposition of Nonlinear Implicit Differential Equations

In engineering, as well as in physics, one often encounters implicit systems of ordinary differential equations of the form Fi(ẏ1…ẏm, y1…ym) = 0, i = 1,…, m, in the unknowns y1…ym, where the Jacobian matrix (∂Fi/∂ẏj)i,j is identically singular. We first state a condition under which the above equation may be solved. Then, we give a decomposition, which provides as a byproduct a clear-cut definition of the index. Since the Fi's must be differentiated an arbitrary number of times, we employ tools stemming from the differential geometry of infinite jets and prolongations like Lie-Bäcklund morphisms.