Ordinary differential equation

About: Ordinary differential equation is a research topic. Over the lifetime, 33149 publications have been published within this topic receiving 590467 citations. The topic is also known as: ODE.

Initial Value Problem for B -Hyperbolic Equation with Integral Condition of the Second Kind

Abstract: For the hyperbolic equation with Bessel operator, we study the initial boundaryvalue problem with integral nonlocal condition of the second kind in a rectangular domain. The integral identity method is used to prove the uniqueness of the solution to the posed problem. The solution is constructed as a Fourier–Bessel series. To justify the existence of the solution to the nonlocal problem, we obtain sufficient conditions to be imposed on the initial conditions to ensure the convergence of the constructed series in the class of regular solutions.

A new implicit six-step P-stable method for the numerical solution of Schrödinger equation

Abstract: In this paper, we will present a new six-step P-stable method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical integration of the one-dimensional Schrodinger equation. We perform an analysis of the local truncation error of the methods for the general case and for the special case of the Schrodinger equation, where we show the decrease of the maximum power of the energy in relation to the corresponding classical methods. We also perform a periodicity analysis. In addition, we determine their periodicity regions. Finally, we compare the new methods to the corresponding classical ones and other known methods from the literature, where we show the high efficiency of the new methods.

Integrability of geodesic motions in curved manifolds through nonlocal conserved charges

Abstract: In this work we study the general system of geodesic equations for the case of a massive particle moving on an arbitrary curved manifold. The investigation is carried out from the symmetry perspective. By exploiting the parametrization invariance property of the system we define nonlocal conserved charges that are independent from the typical integrals of motion constructed out of possible Killing vectors/tensors of the background metric. We show that with their help every two-dimensional surface can---at least in principle---be characterized as integrable. Due to the nonlocal nature of these quantities no more than two can be used at the same time unless the solution of the system is known. We demonstrate that even so, the two-dimensional geodesic problem can always be reduced to a single first order ordinary differential equation; we also provide several examples of this process.

Multidomain spectral method for the Gauss hypergeometric function

Abstract: We present a multidomain spectral approach for Fuchsian ordinary differential equations in the particular case of the hypergeometric equation. Our hybrid approach uses Frobenius’ method and Moebius transformations in the vicinity of each of the singular points of the hypergeometric equation, which leads to a natural decomposition of the real axis into domains. In each domain, solutions to the hypergeometric equation are constructed via the well-conditioned ultraspherical spectral method. The solutions are matched at the domain boundaries to lead to a solution which is analytic on the whole compactified real line $\mathbb {R}\cup {\infty }$, except for the singular points and cuts of the Riemann surface on which the solution is defined. The solution is further extended to the whole Riemann sphere by using the same approach for ellipses enclosing the singularities. The hypergeometric equation is solved on the ellipses with the boundary data from the real axis. This solution is continued as a harmonic function to the interior of the disk by solving the Laplace equation in polar coordinates with an optimal complexity Fourier–ultraspherical spectral method. In cases where logarithms appear in the solution, a hybrid approach involving an analytical treatment of the logarithmic terms is applied. We show for several examples that machine precision can be reached for a wide class of parameters, but also discuss almost degenerate cases where this is not possible.

Asymptotic behaviour for a class of non-monotone delay differential systems with applications

Abstract: The paper concerns a class of $n$-dimensional non-autonomous delay differential equations obtained by adding a non-monotone delayed perturbation to a linear homogeneous cooperative system of ordinary differential equations. This family covers a wide set of models used in structured population dynamics. By exploiting the stability and the monotone character of the linear ODE, we establish sufficient conditions for both the extinction of all the populations and the permanence of the system. In the case of DDEs with autonomous coefficients (but possible time-varying delays), sharp results are obtained, even in the case of a reducible community matrix. As a sub-product, our results improve some criteria for autonomous systems published in recent literature. As an important illustration, the extinction, persistence and permanence of a non-autonomous Nicholson system with patch structure and multiple time-dependent delays are analysed.