Cauchy problem for a second-order ordinary differential equation with a continual derivative

For a second order linear ordinary differential equation with a continual derivative, we construct a fundamental solution. By using the fundamental solution, we find the solution of the Cauchy problem for the considered equation.     

Method of solving the Cauchy problem for one-dimensional polywave equation with singular Bessel operator

We study the Cauchy problem for an equation with singular Bessel operator. Unlike traditional methods to solve this problem, we apply Erde´ lyi–Kober fractional operator and find an explicit formula for the desired solution. We prove that the resulting formula is a unique classical solution to the problem.     

Matrix methods for solving the boundary-value problem for a second-order differential equation with delayed argument

Zero-rank matrix numerical differentiation algorithms are applied to construct efficient numerical-analytical methods (so-called zero-rank matrix methods) to find the eigenvalues and eigenfunctions of boundary-value problems for second-order differential equations with a delayed argument. The proposed methods are analyzed.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 19–24, 1986.     

An operator method of solving the Cauchy problem for a homogeneous system of partial differential equations

On the basis of a generalized separation-of-variables method we propose an operator method of constructing the solution of the Cauchy problem for a homogeneous system of partial differential equations of first order with respect to time and of infinite order with respect to the spatial variables. Translated fromMatematichni Metody i Fiziko-Mekhanichni Polya, Vol. 38, 1995.     

A certain ill-posed Cauchy problem for first-order and second-order differential equations with operator coefficients

Translated from Sibirskii Matematicheskii, Vol. 35, No. 3, pp. 702–706, May–June, 1994.     

Cauchy problem for a differential equation in the Banach space with generalized strongly positive operator coefficient

The concept of strongly positive operator is generalized and the properties of the introduced operators are analyzed. The solutions of the Cauchy problem for a linear inhomogeneous differential equation with generalized strongly positive operator coefficient are obtained.     

Initial-value problem for a continuous second-order differential equation

We obtain necessary initial conditions for a continuous second-order differential equation.     

Cauchy problem for a second-order hyperbolic operator-differential equation with a singular coefficient

In a Hilbert space, we study the well-posedness of the Cauchy problem for a second-order operator-differential equation with a singular coefficient.     

On one method for the analysis of the Cauchy problem for a singularly perturbed inhomogeneous second-order linear differential equation

A sequence converging to the solution of the Cauchy problem for a singularly perturbed inhomogeneous second-order linear differential equation is constructed. This sequence is also asymptotic in the sense that the deviation (in the norm of the space of continuous functions) of its nth element from the solution of the problem is proportional to the (n + 1)th power of the perturbation parameter. A similar sequence is constructed for the case of an inhomogeneous first-order linear equation, on the example of which the application of such a sequence to the justification of the asymptotics obtained by the method of boundary functions is demonstrated.     

Cauchy problem for heat-transfer equation with irregular elliptic operator

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 5, pp. 584–590, May, 1989.