Separation of variables in the hydrodynamic stability equations

Stability of some unsteady three-dimensional flows (exact solutions of the viscous incompressible Navier–Stokes equations in cylindrical coordinates) is studied via separation of variables in the linearized equations for the flow perturbations. The flows in an expanding rotating porous cylinder and in a gap between two coaxial rotating cylinders are considered. Converting the stability equations to the new variables allows perturbation forms (counterparts of normal modes of the steady state parallel flow stability problem) such that the linear stability problems are exactly reduced to eigenvalue problems of ordinary differential equations. The eigenvalue problems are solved numerically with the help of the spectral collocation method based on Chebyshev polynomials. The results showing dependence of the stability threshold on the parameters of the problems and a spatial structure of the unstable perturbation modes are presented. For some classes of perturbations, exact analytical solutions of the eigenvalue problems are available. A combination of analytical and numerical solutions can provide useful testing for numerical methods used in the hydrodynamic stability studies. It may also provide a basis for a well-grounded discussion of some problematic points of (numerical) stability analysis. In particular, in the present paper, a problem of formulation of the boundary conditions for perturbations at the axis r=0 is discussed on the basis of the solutions obtained.

The Green’s function is widely used in solving boundary value problems for differential equations, to which many mathematical and physical problems are reduced. In particular, solutions of partial differential equations by the Fourier method are reduced to boundary value problems for ordinary differential equations. Using the Green's function for a homogeneous problem, one can calculate the solution of an inhomogeneous differential equation. Knowing the Green's function makes it possible to solve a whole class of problems of finding eigenvalues in quantum field theory. The developed method for constructing the Green’s function of boundary value problems for ordinary linear differential equations is described. An algorithm and program for calculating the Green's function of boundary value problems for differential equations of the second and third orders in an explicit analytical form are presented. Examples of computing the Green's function for specific boundary value problems are given. The fundamental system of solutions of ordinary differential equations with singular points needed to construct the Green's function is calculated in the form of generalized power series with the help of the developed programs in the Maple environment. An algorithm is developed for constructing the Green's function in the form of power series for second-order and third-order differential equations with given boundary conditions. Compiled work programs in the Maple environment for calculating the Green functions of arbitrary boundary value problems for differential equations of the second and third orders. Calculations of the Green's function for specific third-order boundary value problems using the developed program are presented. The obtained approximate Green’s function is compared with the known expressions of the exact Green’s function and very good agreement is found

On the variational iteration method and other iterative techniques for nonlinear differential equations

“The last decade has seen such a deluge of papers on the numerical solution of the initial value problems for ordinary differential equations that it is quite impracticable to list, far less to summarize, all the contributions.” These are the opening words of J. D. Lambert’s paper [54] in the State of the Art 1976 Conference Proceedings. In spite of such a disclaimer, the paper succeeds in presenting, in less than fifty large-print pages, a unified view of all that, at the time of its writing, was known on numerical initial-value problems (IVP) for ordinary differential equations (ODEs). Ten years later, the subject was just too big and although the State of the Art 1986 Conference [50] featured three ODE speakers (J.D. Lambert, A.R. Curtis and G. Wanner), their combined contributions are very far from surveying all the numerical ODE field.

The difficulties arising in the linear theory of viscous fluid flow stability are to a great extent connected with the mathematical problems of finding eigenvalues of non-self-conjugated differential operators. Finding the eigenvalues of the Orr-Sommerfeld spectral problem is a difficult problem of computational mathematics. The methodology of applying the method of regularized traces for finding eigenvalues of the classical spectral problem of the Orr-Sommerfeld hydrodynamic stability theory is considered in this paper. A new method for approximate computation of the first eigenvalues of spectral problems generated by discrete semi-limited operators has been developed. This method is not iterative in contrast to the known numerical methods for finding eigenvalues of operators. New estimates of the regularized traces of the discrete operator have been obtained in this work. The formulas by which it is possible to calculate the corrections of the necessary order perturbation theory have been found. The computational experiment to calculate the first eigenvalues of the Orr-Sommerfeld spectral problem has been performed.

Furthering research on linear multistep hybrid block methods is essential to enhance accuracy, stability, and efficiency in solving ordinary differential equations, enabling advanced modelling of complex systems across science and engineering for better predictive analysis and real-world applications. In this study, an efficient linear multistep hybrid block method with a single-step and seven off-step points for the direct numerical integration of fifth-order initial value problems (IVPs) in ordinary differential equations (ODEs), eliminating the need for reduction to a system of first-order ODEs is proposed. The method is constructed using a collocation approach at both grid and off-grid points, alongside interpolation at five off-grid points, to approximate the solution via a power series polynomial. The resulting system of equations is solved to obtain the necessary discrete and additional formulae that constitute the block approach. A comprehensive theoretical analysis confirms that the method possesses desirable numerical properties, including a well-defined order, zero stability, consistency, convergence, and absolute stability. Comparative numerical experiments against existing methods demonstrate that the proposed approach achieves superior accuracy and efficiency, making it a promising tool for solving both linear and nonlinear fifth-order ODEs.

The problem of the stability of nonplane-parallel flows is one of the most difficult and least studied problems in the theory of hydrodynamic stability [1]. In contrast to the Heisenberg approximation [1], the basic state whose stability is investigated depends on several variables, and the stability problem reduces to the solution of an eigenvalue problem for partial differential equations in which the coefficients depend on several variables [2–7]. In the case of a periodic dependence of these coefficients on the time [2] or the spatial coordinates [3, 4], the analog of Floquet theory for the partial differential equations is constructed. With rare exceptions, the case of a nonperiodic dependence has usually been considered under the assumption of weak nonplane-parallelism, i.e., a fairly small deviation from the plane-parallel case has been assumed and the corresponding asymptotic expansions in the linear [6] and nonlinear [7] stability analyses considered. The present paper considers the case of an arbitrary dependence of the velocity profile of the basic flow on two spatial variables. The deviation from the plane-parallel case is not assumed to be small, and the corresponding eigenvalue problem for the partial differential equations is solved by means of the direct methods of [5], which were introduced for the first time and justified in the theory of hydrodynamic stability by Petrov [8].

Nonlinear Boundary Value Problems for Ordinary Differential Equations: From Schauder Theorem to Stable Homotopy

A method for constructing a solution for some systems of first-order quasilinear partial differential equations is presented. The type of equations can either be hyperbolic or elliptic. The method is based on the application of the generalized hodograph method, which allows us to write the solution in an implicit form. There is a system of first-order linear partial differential equations that is used for commuting flows in the generalized hodograph method. We discover an analogy between the commuting flows and divided differences for the Hermite polynomial. This analogy allows us to obtain an explicit representation for commuting flows. The introduction of new (Lagrangian) variables, which are conserved on the characteristics of the original system, suggests a way to transform the solution of the Cauchy problem for first-order quasilinear partial differential equations to the solution of the Cauchy problem for ordinary differential equations. Numerical, and in some cases analytical, integration of the Cauchy problem makes it possible to construct explicit solutions of the problem on the level lines (isochrons) of the implicit solution. The method proposed is significantly different from the grid method, finite element method, finite volume method, and, in fact, is more precise. The error of the solution can arise only at the last stage in the numerical integration of the Cauchy problem for ordinary differential equations. Moreover, the method allows us to obtain multivalued solutions, in particular, to study the process of wave breaking in hyperbolic systems. Particular cases of the equations considered here describe diffusion-free approximation in a wide range of mass transport processes in multicomponent mixtures, such as electrophoresis, chromatography, centrifugation. As a simple example, the solution of the electrophoresis problem (separation multicomponent mixture to individual component) is presented.KeywordsGeneralized hodograph methodCommuting flowsDivided differenceMathematical Subject Classification (2000)35L4535L4035L65

A new one-step block method with generalized three hybrid points for solving initial value problems of second-order ordinary differential equations directly is proposed. In deriving this method, a power series approximate function is interpolated at {xn,xn+r} while its second and third derivatives are collocated at all points {xn,xn+r,xn+s,xn+t,xn+1} in the given interval. The proposed method is then tested on initial value problems of second-order ordinary differential equations solved by other methods previously. The numerical results confirm the superiority of the new method to the existing methods in terms of accuracy.

The application of block methods in the study of dynamical system is one area which needs more research. Therefore, this work presents block methods and its application to solve some problems in solid and fluid mechanics. The method was developed directly using the approaches of collocation and interpolation, and using power series polynomial as a trial solution. First, the system of linear equations is solved to obtain the unknown coefficients. The coefficients gotten are then substituted into the approximate solution to obtain continuous scheme. The continuous scheme, its first, second and third derivatives are evaluated at all the grid points to generate the block methods. The derived methods were applied to solve fourth-order boundary value problems of ordinary differential equations arising from beams and chemical problems. The results demonstrate the reliability and efficiency of the proposed method.

In this paper, we present the theory of nonstandard finite difference schemes which can be used to solve some initial value problems in ordinary differential equations. Methods of construction and mode of implementation of these methods are also discussed. We also examine the stability properties of the constructed schemes.

Singular perturbed problems in ordinary differential equations

A posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations

Approximating functions in spectral methods are related to polynomial solutions of eigenvalue problems in ordinary differential equations, known as Sturm-Liouville problems. These originate from applying the method of separation of variables in the analysis of boundary-value problems. We outline both basic and remarkable properties of the most commonly used families of polynomials of this kind.