First-order Quasilinear Differential Equations Research Articles

On a Certain First-Order Differential Equation with Delay

We consider the Cauchy problem for a first-order quasilinear differential equation with delayed argument of neutral type, and obtain sufficient conditions of existence and uniqueness of its solutions. Proofs of the solvability of nonlinear problems, estimates of solutions, and constructions of approximate methods are based on Chaplygin-type theorems on differential inequalities.

On the unique continuation of the germs of solutions of first-order differential equations along curves

For solutions of linear, weakly nonlinear, and quasilinear first-order differential equations, we obtain theorems on the unique unbounded continuation of germs along continuous curves contained in the integral submanifolds of the distribution induced by the major part of the equation.

Integrating rotation and angular velocity from curvature

The problem of integrating the rotational vector from a given angular velocity vector is met in such diverse fields as the navigation, robotics, computer graphics, optical tracking and non-linear dynamics of flexible beams. For example, if the numerical formulation of non-linear dynamics of flexible beams is based on the interpolation of curvature, one needs to derive the rotation from the assumed curvature field. The relation between the angular velocity and the rotation is described by the first-order quasi-linear differential equation. If the rotation is given, the related angular velocity is obtained by the differentiation. By contrast, if the angular velocity is given, the related rotations are obtained by the integration. The exact closed-form solution for the rotation is only possible if the angular velocity is constant in time. In dynamics of non-linear flexible spatial beams, the problem of integrating rotations from a given angular velocity becomes even more complex because both the angular velocity and the curvature need simultaneously be integrated and are both functions of space and time. As the angular velocity and the curvature are assumed to be analytic functions, they must satisfy certain integrability conditions to assure the unique rotation is obtained from the two differential equations. The objective of the present paper is to derive approximate, yet closed-form solutions of the following problem: for a given curvature vector, determine both the rotation and the angular velocity. In order to avoid the singularity of kinematic relations, the quaternions are used for the parametrization of rotations, and the integrations are partly performed in the four-dimensional quaternion space. The resulting closed-form expressions for the rotational and angular velocity quaternions are ready to be used in the finite-element formulations of the dynamics of flexible spatial beams as interpolating functions. The present novel solution is assessed by comparisons of the numerical results with analytical solutions for variety of oscillating curvature functions, as well as with the solutions of the quaternion-based midpoint integrator and the Runge–Kutta-based Crouch–Grossman geometrical methods CG3 and CG4.

An Optimal Control Problem for Quasilinear Differential Equations with Bitsadze–Samarski Boundary Conditions

The present paper is devoted to optimal control problems whose behavior is described by quasilinear first-order differential equations on the plane with nonlocal Bitsadze–Samarski boundary conditions. A theorem on the existence and uniqueness of a generalized solution in the space $$ {C}_{\mu}\left(\overline{G}\right) $$ is proved for quasilinear differential equations; necessary optimality conditions are obtained in terms of the maximum principle; the Bitsadze–Samarski boundary-value problem is examined for a first-order linear differential equation; the existence of a solution in the space $$ {C}_{\mu}^p\left(\overline{G}\right) $$ is proved, and an a priori estimate is derived. A necessary and sufficient optimality condition is proved for a linear optimal control problem.