Nemlineáris peremérték-feladatok megoldásainak vizsgálata = Investigation of solutions of nonlinear boundary-value problems

A pályázat általános alakú nemlineáris közönséges differenciálegyenlet-rendszerekhez illetve lineáris funkcionál- differenciálegyenlet rendszerekhez rendelt peremérték-feladatok megoldásainak vizsgálatával foglalkozik. A fő hangsúlyt lokálisan Lipschitz-rendszerekre helyezzük. Kutatásaink kimutatták, hogy a sorozatos közelítésen alapuló saját fejlesztésű numerikus –analitikus módszereink az egzisztencia vizsgálatban illetve a megoldás közelítő meghatározása során felhasználhatók akkor is, amikor más csoportokhoz tartozó módszerek alkalmazhatóságát biztosító feltételek nem teljesülnek. Kidolgoztunk egy paraméterezési technikát, amely segítségével egyszerűbben tudjuk kezelni a szinguláris mátrixokat tartalmazó két és három pontos lineáris peremfeltételeket. A szóban forgó paraméterezési eljárás igen hasznosnak bizonyult erősen nemlineáris feladatoknál is, amikor az adott peremfeltételek szintén nemlineárisak. A módszerünk segítségével a nemlineáris kétpontos vagy hárompontos peremfeltételek visszavezethetőek egy egyszerűbb feladatra, ahol a transzformált nemlineáris differenciálegyenlet-rendszert már kétpontos lineáris peremfeltételekkel kell tanulmányozni. Ezen kívül a módosított feladathoz egy algebrai (vagy transzcendens) egyenletrendszer egyszeri megoldása tartozik, melynek dimenziója megegyezik a bevezetett paraméterek számával. Periodikus peremfeltételek mellett nem-autonóm differenciálegyenlet- rendszerek esetén kimutattuk a megoldások bizonyos általánosított szimmetrikus tulajdonságait Ez a szimmetria speciális esetként magába foglalja a páros, páratlan és más ismert tulajdonságokat. | The Project deals with the investigation of solutions of non-linear boundary- value problems for systems of ordinary differential and linear functional-differential equations. We pay the main attention to the local Lipschitz systems. Our investigations show, that the numerical-analytic methods based upon successive approximations developed in the framefork of the project, can be used in the existence analysis and approximate construction of the solutions, even in those cases when the sufficient conditions of the applicability of the methods from other areas are not satisfied. We introduce a suitable parametrization technique and show how it can help when dealing with non-separated three-point boundary conditions determined by singular matrices. This parametrization approach is very usefull in the case of strongly nonlinear problems even if the given boundary conditions are nonlinear. We show that in the investigation such problem, it is often useful to reduce it to a parametrized family of two-point boundary value problems with linear boundary conditions for a suitably perturbed differential systems. Our technique leads to a certain system of algebraic equations for the introduced parameters whose solutions provide those numerical values of the parameters that correspond to the solutions of the given boundary value problem. For non-autonomous non-linear systems of differential equations some new generalized symmetric properties of periodic solutions were determined. The odd, even and some other properties appear as special cases

A new approach for solving nonlinear Thomas-Fermi equation based on fractional order of rational Bessel functions

In this paper, the fractional order of rational Bessel functions collocation method (FRBC) to solve Thomas-Fermi equation which is defined in the semi-infinite domain and has singularity at $x = 0$ and its boundary condition occurs at infinity, have been introduced. We solve the problem on semi-infinite domain without any domain truncation or transformation of the domain of the problem to a finite domain. This approach at first, obtains a sequence of linear differential equations by using the quasilinearization method (QLM), then at each iteration solves it by FRBC method. To illustrate the reliability of this work, we compare the numerical results of the present method with some well-known results in other to show that the new method is accurate, efficient and applicable

Nonlinear equations for p-adic open, closed, and open-closed strings

We investigate the structure of solutions of boundary value problems for a one-dimensional nonlinear system of pseudodifferential equations describing the dynamics (rolling) of p-adic open, closed, and open-closed strings for a scalar tachyon field using the method of successive approximations. For an open-closed string, we prove that the method converges for odd values of p of the form p=4n+1 under the condition that the solution for the closed string is known. For p=2, we discuss the questions of the existence and the nonexistence of solutions of boundary value problems and indicate the possibility of discontinuous solutions appearing.Comment: 16 pages, 3 figure

A machine learning framework for data driven acceleration of computations of differential equations

We propose a machine learning framework to accelerate numerical computations of time-dependent ODEs and PDEs. Our method is based on recasting (generalizations of) existing numerical methods as artificial neural networks, with a set of trainable parameters. These parameters are determined in an offline training process by (approximately) minimizing suitable (possibly non-convex) loss functions by (stochastic) gradient descent methods. The proposed algorithm is designed to be always consistent with the underlying differential equation. Numerical experiments involving both linear and non-linear ODE and PDE model problems demonstrate a significant gain in computational efficiency over standard numerical methods

Solution of 3-dimensional time-dependent viscous flows. Part 2: Development of the computer code

There is considerable interest in developing a numerical scheme for solving the time dependent viscous compressible three dimensional flow equations to aid in the design of helicopter rotors. The development of a computer code to solve a three dimensional unsteady approximate form of the Navier-Stokes equations employing a linearized block emplicit technique in conjunction with a QR operator scheme is described. Results of calculations of several Cartesian test cases are presented. The computer code can be applied to more complex flow fields such as these encountered on rotating airfoils

An approximate solution of the MHD Falkner-Skan flow by Hermite functions pseudospectral method

Based on a new approximation method, namely pseudospectral method, a solution for the three order nonlinear ordinary differential laminar boundary layer Falkner-Skan equation has been obtained on the semi-infinite domain. The proposed approach is equipped by the orthogonal Hermite functions that have perfect properties to achieve this goal. This method solves the problem on the semi-infinite domain without truncating it to a finite domain and transforming domain of the problem to a finite domain. In addition, this method reduces solution of the problem to solution of a system of algebraic equations. We also present the comparison of this work with numerical results and show that the present method is applicable.Comment: 15 pages, 4 figures; Published online in the journal of "Communications in Nonlinear Science and Numerical Simulation

The GNAT method for nonlinear model reduction: effective implementation and application to computational fluid dynamics and turbulent flows

The Gauss--Newton with approximated tensors (GNAT) method is a nonlinear model reduction method that operates on fully discretized computational models. It achieves dimension reduction by a Petrov--Galerkin projection associated with residual minimization; it delivers computational efficency by a hyper-reduction procedure based on the `gappy POD' technique. Originally presented in Ref. [1], where it was applied to implicit nonlinear structural-dynamics models, this method is further developed here and applied to the solution of a benchmark turbulent viscous flow problem. To begin, this paper develops global state-space error bounds that justify the method's design and highlight its advantages in terms of minimizing components of these error bounds. Next, the paper introduces a `sample mesh' concept that enables a distributed, computationally efficient implementation of the GNAT method in finite-volume-based computational-fluid-dynamics (CFD) codes. The suitability of GNAT for parameterized problems is highlighted with the solution of an academic problem featuring moving discontinuities. Finally, the capability of this method to reduce by orders of magnitude the core-hours required for large-scale CFD computations, while preserving accuracy, is demonstrated with the simulation of turbulent flow over the Ahmed body. For an instance of this benchmark problem with over 17 million degrees of freedom, GNAT outperforms several other nonlinear model-reduction methods, reduces the required computational resources by more than two orders of magnitude, and delivers a solution that differs by less than 1% from its high-dimensional counterpart