Cauchy Problem For Ordinary Differential Equations Research Articles

Well-posedness of problem with parameter for an integro-differential equation

AbstractA problem with parameter for an integro-differential equation is approximated by a problem with parameter for a loaded differential equation. The well-posedness of a problem with parameter for the integro-differential equation is established in the terms of the well-posedness of a problem with parameter for the loaded differential equation. A mutual relationship between the qualitative properties of original and approximate problems is obtained, and the estimates for differences between their solutions are set. A new general solution to the loaded differential equation with parameter is presented, and its properties are described. The problem with parameter for the loaded differential equation is reduced to a system of linear algebraic equations with respect to the arbitrary vectors of a general solution introduced. The system’s coefficients and right-hand sides are computed by solving the Cauchy problems for ordinary differential equations.

Sobolev-orthogonal systems of functions and the Cauchy problem for ODEs

We consider systems of functions (, ) that are Sobolev-orthonormal with respect to a scalar product of the form and are generated by a given orthonormal system of functions (). The Fourier series and sums with respect to the system (, ) are shown to be a convenient and efficient tool for the approximate solution of the Cauchy problem for ordinary differential equations (ODEs).

A new semi-analytical approach for numerical solving of cauchy problem for differential equations with delay

In the paper, we present new semi-analytical approach for FDE?s consisting in combination of the method of steps and a technique called differential transformation method (DTM). This approach reduces the original Cauchy problem for delayed or neutral differential equation to Cauchy problem for ordinary differential equation for which DTM is convenient and efficient method. Moreover, there is no need of any symbolic calculations or initial approximation guesstimates in contrast to methods like the homotopy analysis method, the homotopy perturbation method, the variational iteration method or the Adomian decomposition method. The efficiency of the proposed method is shown on certain classes of FDE?s with multiple constant delays including FDE of neutral type. We also compare it to the current approach of using DTM and the Adomian decomposition method where Cauchy problem is not well posed.