Implicit System Of Differential Equations Research Articles

On a Control Problem for a System of Implicit Differential Equations

We consider the differential inclusion F(t,x,x˙)∋0 with the constraint x˙(t)∈B(t), t∈[a,b], on the derivative of the unknown function, where F and B are set-valued mappings, F:[a,b]×Rn×Rn×Rm⇉ is superpositionally measurable, and B:[a,b]⇉Rn is measurable. In terms of the properties of ordered covering and the monotonicity of set-valued mappings acting in finite-dimensional spaces, for the Cauchy problem we obtain conditions for the existence and estimates of solutions as well as conditions for the existence of a solution with the smallest derivative. Based on these results, we study a control system of the form f(t,x,x˙,u)=0, x˙(t)∈B(t), u(t)∈U(t,x,x˙), t∈[a,b].

On a Control Problem for a System of Implicit Differential Equations

On a Control Problem for a System of Implicit Differential Equations

On a Riemann–Liouville Type Implicit Coupled System via Generalized Boundary Conditions

We study a coupled system of implicit differential equations with fractional-order differential boundary conditions and the Riemann–Liouville derivative. The existence, uniqueness, and at least one solution are established by applying the Banach contraction and Leray–Schauder fixed point theorem. Furthermore, Hyers–Ulam type stabilities are discussed. An example is presented to illustrate our main result. The suggested system is the generalization of fourth-order ordinary differential equations with anti-periodic, classical, and initial boundary conditions.

О задаче Коши для неявных дифференциальных уравнений высших порядков

The article is devoted to the study of implicit differential equations based on statements about covering mappings of products of metric spaces. First, we consider the system of equations Φ_i (x_i,x_1,x_2,…,x_n )=y_i, i=(1,n,) ̅ where 〖 Φ〗_i: X_i×X_1×… ×X_n→Y_i, y_i∈Y_i, X_i and Y_i are metric spaces, i=(1,n) ̅. It is assumed that the mapping 〖 Φ〗_i is covering in the first argument and Lipschitz in each of the other arguments starting from the second one. Conditions for the solvability of this system and estimates for the distance from an arbitrary given element x_0∈X to the set of solutions are obtained. Next, we obtain an assertion about the action of the Nemytskii operator in spaces of summable functions and establish the relationship between the covering properties of the Nemytskii operator and the covering of the function that generates it. The listed results are applied to the study of a system of implicit differential equations, for which a statement about the local solvability of the Cauchy problem with constraints on the derivative of a solution is proved. Such problems arise, in particular, in models of controlled systems. In the final part of the article, a differential equation of the n-th order not resolved with respect to the highest derivative is studied by similar methods. Conditions for the existence of a solution to the Cauchy problem are obtained.

A fully Hadamard and Erdélyi–Kober-type integral boundary value problem of a coupled system of implicit differential equations

In this article, we give sufficient conditions for the existence of solutions for a new coupled system of second-order implicit differential equations with Hadamard and Erdelyi-Kober fractional integral boundary conditions and nonlocal conditions at the boundaries in Banach space. The main result is based on a Monch fixed point theorem combined with the measure of noncompactness of Kuratowski; an example is given to illustrate our approach.

A Hamilton–Jacobi theory for implicit differential systems

In this paper, we propose a geometric Hamilton-Jacobi theory for systems of implicit differential equations. In particular, we are interested in implicit Hamiltonian systems, described in terms of Lagrangian submanifolds of TT*Q generated by Morse families. The implicit character implies the nonexistence of a Hamiltonian function describing the dynamics. This fact is here amended by a generating family of Morse functions which plays the role of a Hamiltonian. A Hamilton–Jacobi equation is obtained with the aid of this generating family of functions. To conclude, we apply our results to singular Lagrangians by employing the construction of special symplectic structures.

НЕПРЕРЫВНАЯ ЗАВИСИМОСТЬ ОТ ПАРАМЕТРОВ РЕШЕНИЯ КРАЕВОЙ ЗАДАЧИ ДЛЯ СИСТЕМЫ НЕЯВНЫХ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ С ОТКЛОНЯЮЩИМСЯ АРГУМЕНТОМ

НЕПРЕРЫВНАЯ ЗАВИСИМОСТЬ ОТ ПАРАМЕТРОВ РЕШЕНИЯ КРАЕВОЙ ЗАДАЧИ ДЛЯ СИСТЕМЫ НЕЯВНЫХ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ С ОТКЛОНЯЮЩИМСЯ АРГУМЕНТОМ

О ЗАДАЧЕ КОШИ ДЛЯ СИСТЕМ НЕЯВНЫХ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ

О ЗАДАЧЕ КОШИ ДЛЯ СИСТЕМ НЕЯВНЫХ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ

Корректная разрешимость задач управления для систем дифференциальных уравнений неявного вида

Корректная разрешимость задач управления для систем дифференциальных уравнений неявного вида

Mathematical model of pulsatile flow of non-Newtonian fluid in tubes of varying cross-sections and its implications to blood flow

The effects of rheological behavior of blood and pulsatility on flow through an artery with stenosis have been investigated. Blood has been represented by a non-Newtonian fluid obeying Herschel–Bulkley equation. Using the Reynolds number as the perturbation parameter, a perturbation technique is adopted to solve the resulting quasi-steady non-linear coupled implicit system of differential equations. Analytical expressions for velocity distribution, wall shear stress, volumetric flow rate and the mean flow resistance have been obtained. It is observed that the wall shear stress and flow resistance increase for increasing value of yield stress with other parameters held fixed. One of the remarkable results of the present analysis is not only to bring out the effect of the size of the stenosis but also to study the influence of the shape of the stenosis. The change in the shape of the stenosis brings out a significant change in the value of flow resistance but it has no effect on the variation of wall shear stress except shifting the point (where it attains its maximum value) towards downstream. It is pertinent to point out that pulsatile flow of Newtonian fluid, Bingham plastic fluid and Power-law fluid become particular cases of the present model. The present approach has general validity in comparison with many mathematical models developed by others and may be applied to any mathematical model by taking into account of any type of rheological property of blood. The obtained velocity profiles have been compared with the experimental data and it is observed that blood behaves like a Herschel–Bulkley fluid rather than Power-law and Bingham fluids. Finally, some biorheological applications of the present model have briefly been discussed.

On the topological classification of rarefaction curves in systems of three conservation laws

In this paper we study the topological properties of integral curves of a system of implicit differential equations associated to rarefaction curves of a system of three conservation laws. This system of equations becomes singular at the points of eigenvalue of multiplicity greater or equal to two. We focus our attention to the generic case of multiplicity two and three. We give local weak topological models for these equations.

On the index and the order of quasi-regular implicit systems of differential equations

This paper is mainly devoted to the study of the differentiation index and the order for quasi-regular implicit ordinary differential algebraic equation (DAE) systems. We give an algebraic definition of the differentiation index and prove a Jacobi-type upper bound for the sum of the order and the differentiation index. Our techniques also enable us to obtain an alternative proof of a combinatorial bound proposed by Jacobi for the order. As a consequence of our approach we deduce an upper bound for the Hilbert–Kolchin regularity and an effective ideal membership test for quasi-regular implicit systems. Finally, we prove a theorem of existence and uniqueness of solutions for implicit differential systems.

Legendre singularities in systems of implicit ODEs and slow-fast dynamical systems

Generic properties of regular first integrals of systems of implicit differential equations are considered. In particular, for systems of two equations with two phase variables, a classification of generic bifurcations of integral level surfaces is described.

Multidimensional Poincaré construction and singularities of lifted fields for implicit differential equations

This paper is devoted to singular points of the so-called lifted vector fields, which arise in studying systems of implicit differential equations by using the method of lifting the equation to a surface, a generalization of the construction used by Poincare for a single implicit equation. The author studies the phase portraits and renormal forms of such fields in a neighborhood of their singular points. In conclusion, this paper considers the lifted vectors fields generated by Euler-Lagrange and Euler-Poisson equations and fast-slow systems.

An explicit solution to a system of implicit differential equations

In this article we give an explicit solution to the vectorial differential inclusion where we have to find a map u:Ω⊂R2→R2 with u=0 on ∂Ω andDu(x)∈O(2),a.e. x∈Ω.

Pulsatile flow of Herschel–Bulkley fluid through catheterized arteries – A mathematical model

The pulsatile flow of blood through catheterized artery has been studied in this paper by modeling blood as Herschel–Bulkley fluid and the catheter and artery as rigid coaxial circular cylinders. The Herschel–Bulkley fluid has two parameters, the yield stress θ and the power index n. Perturbation method is used to solve the resulting quasi-steady nonlinear coupled implicit system of differential equations. The effects of catheterization and non-Newtonian nature of blood on yield plane locations, velocity, flow rate, wall shear stress and longitudinal impedance of the artery are discussed. The existence of two yield plane locations is investigated and their dependence on yield stress θ, amplitude A, and time t are analyzed. The width of the plug core region increases with increasing value of yield stress at any time. The velocity and flow rate decrease, whereas wall shear stress and longitudinal impedance increase for increasing value of yield stress with other parameters held fixed. On the other hand, the velocity, flow rate and wall shear stress decrease but resistance to flow increases as the catheter radius ratio (ratio of catheter radius to vessel radius) increases with other parameters fixed. The results for power law fluid, Newtonian fluid and Bingham fluid are obtained as special cases from this model.

On implicit systems of differential equations

This article considers implicit systems of differential equations. The implicit systems that are considered are given by polynomial relations on the coordinates of the indeterminate function and the coordinates of the time derivative of the indeterminate function. For such implicit systems of differential equations, we are concerned with computing algebraic constraints such that on the algebraic variety determined by the constraint equations the original implicit system of differential equations has an explicit representation. Our approach is algebraic. Although there have been a number of articles that approach implicit differential equations algebraically, all such approaches have relied heavily on linear algebra. The approach of this article is different, we have no linearity requirements at all, instead we rely on algebraic geometry. In particular, we use birational mappings to produce an explicit system. The methods developed in this article are easily implemented using various computer algebra systems.

Estimation of increased flow resistance in a narrow catheterized artery — a theoretical model

The changed flow pattern in a narrow catheterized artery is studied and an estimate of the increased flow resistance is made. The anomalous behaviour of blood in small blood vessels has been taken into account by modelling blood as a Casson fluid possessing some finite yield stress. Both the cases of steady and pulsatile flow situations are studied. The pulsatile flow is analysed by considering the pressure gradient as a periodic function of time with small inertial effects. The resulting quasi-steady non-linear coupled implicit system of differential equations governing the flow are solved using a perturbation analysis, where it is assumed that the Womersley frequency parameter is small (α < 1) which is reasonable for physiological situations in small blood vessels as well as in coronary arteries. The effect of pulsatility, catheter radius and yield stress of the fluid on the yield plane locations, velocity distribution, flow rate, shear stress and frictional resistance are investigated. Because of the yield stress θ, two yield surfaces are found to be located in the flow field. Depending on the ratio k (catheter size/vessel size) ranging from 0.3 to 0.7 (which is widely used in coronary angioplasty procedures), the frictional resistance to flow in large blood vessels, where the effect of yield stress can be neglected (i.e. θ = 0), increases by a factor ranging from 3 to 33. In small blood vessels with the same range of catheter size and an unit pressure gradient, frictional resistance increase was by a factor of 7-21 when θ = 0.05 and 11-294 when θ = 0.1. For small values of k and θ, the frictional resistance increase was relatively less, but still appreciable. For comparative large values of k and θ, the frictional resistance increases to several hundred times thus implying that the combined effect of increased catheter radius and yield stress is to obstruct the fluid movement considerably.

Effect of yield stress on the flow of a Casson fluid in a homogeneous porous medium bounded by a circular tube

The effect of yield stress on the flow characteristics of a Casson fluid in a homogeneous porous medium bounded by a circular tube is investigated by employing the Brinkman model to account for the Darcy resistance offered by the porous medium. The non-linear coupled implicit system of differential equations governing the flow is first transformed into suitable integral equations and are solved numerically. Analytical solution is obtained for a Newtonian fluid in the case of constant permeability, and the numerical solution is verified with that of the analytic solution. The effect of yield stress of the fluid and permeability of the porous medium on shear stress and velocity distributions, plug flow radius and flow rate are examined. The minimum pressure gradient required to start the flow is found to be independent of the permeability of the porous medium and is equal to the yield stress of the fluid.

Boundary Value Problems for Systems of Implicit Differential Equations

Boundary Value Problems for Systems of Implicit Differential Equations