Mixed problem for a first-order partial differential equation with involution and periodic boundary conditions

The Fourier method is used to find a classical solution of the mixed problem for a first-order differential equation with involution and periodic boundary conditions. The application of the Fourier method is substantiated using refined asymptotic formulas obtained for the eigenvalues and eigenfunctions of the corresponding spectral problem. The Fourier series representing the formal solution is transformed using certain techniques, and the possibility of its term-by-term differentiation is proved. Minimal requirements are imposed on the initial data of the problem.     

Quasilinearization for the periodic boundary value problem for hybrid differential equation

The method of quasilinearization for a periodic boundary value problem for nonlinear hybrid differential equations is studied. It is shown that the convergence is quadratic.     

Approximate solution to a boundary value problem for the Lyapunov differential equation with a parameter

A special boundary value problem is studied for the Lyapunov differential equation which is used for investigation of the asymptotic properties of solutions to systems of periodic differential equations with a parameter. An algorithm is proposed for constructing an approximate solution to this boundary value problem, and conditions on the parameter are found under which the zero solution to the system is asymptotically stable.     

Exact multiplicity for periodic solutions of a first-order differential equation

Consider the differential equation

(1)     

On a periodic boundary value problem for cyclic feedback type linear functional differential systems

Nonimprovable effective sufficient conditions are established for the unique solvability of the periodic problem

Impulsive periodic boundary value problems of first-order differential equations

In this paper, by using Schaeffer's theorem, we prove new existence theorems for a nonlinear periodic boundary value problem of first-order differential equations with impulses. Our results improve and generalize some known results.     

Mean periodic solutions of a linear inhomogeneous first-order differential equation with random coefficients

We obtain deterministic first-order linear differential equations with ordinary and variational derivatives and deterministic initial conditions for the expectation and the second moment function of the solution of an ordinary scalar first-order linear inhomogeneous differential equation whose coefficients are random processes. We derive existence conditions for mean periodic solutions. In particular, we consider Gaussian and uniformly distributed random coefficients.     

Existence of solution for boundary value problem of nonlinear fractional differential equation

This paper is concerned with the boundary value problem of a nonlinear fractional differential equation.By means of Schauder fixed-point theorem,an existence result of solution is obtained.     

Periodic boundary value problem for the first order functional differential equations with impulses

This paper is concerned with the existence and approximation of solutions for a class of first order impulsive functional differential equations with periodic boundary value conditions. A new comparison result is presented and the previous results are extended.     

Periodic boundary value problem for first-order impulsive ordinary differential equations

This paper investigates the existence of minimal and maximal solutions of the periodic boundary value problem for first-order impulsive differential equations by establishing two comparison results and using the method of upper and lower solutions and the monotone iterative technique.     

Periodic boundary value problem for first-order impulsive functional differential equations

In this paper, by means of the method of upper and lower solutions and the monotone iterative technique, the existence of minimal and maximal solutions of the periodic boundary value problem for first-order impulsive functional differential equations is considered.     

Fundamental solution for linear two-point boundary value problem

The quadratic functional minimization with differential restrictions represented by the command linear systems is considered. The optimal solution determination implies the solving of a linear problem with two points boundary values. The proposed method implies the construction of a fundamental solution S(t)—a n×n matrix—and of a vector h(t) defining an adjoint variable λ(t) depending of the state variable x(t). From the extremum necessary conditions it is obtained the Riccati matrix differential equation having the S(t) as unknown fundamental solution is obtained. The paper analyzes the existence of the Riccati equation solution S(t) and establishes as the optimal solution of the proposed optimum problem. Also a superior limit of the minimum for the considered quadratic functionals class are evaluated.     

On the numerical integration of a boundary value problem involving a fourth order linear differential equation

A finite difference method for the numerical integration of the linear two-point boundary value problem $$y^{(4)} + f(x)y = g(x),y(a) = A_1 ,y(b) = A_2 ,y''(a) = B_1 ,y''(b) = B_2$$ is constructed and analyzed. The method usesf′,f″,g′,g″ at the boundary points to obtain anO(h ⁶) global error, while requiring only the solution of a system of linear equations associated with a five-band matrix. A typical numerical example is included to demonstrate the practical usefulness of the numerical procedure as well.