Linear stability analysis in the numerical solution of initial value problems

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

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

The purpose of this study is to develop a numerical method for bifurcation analysis of nonlinear partial differential equations, based on the reduction of partial differential equations to ordinary ones, using the Kolmogorov-Arnold theorem. Methods. This paper describes a method for reducing partial differential equations to ordinary ones using the Kolmogorov-Arnold theorem, as well as methods for the bifurcation analysis of nonlinear boundary value problems for ordinary differential equations. Results. The paper presents a new method for solving and bifurcation analysis of nonlinear boundary value problems for partial differential equations, which allow variational formulation. The method was applied to a nonlinear two-dimensional Bratu problem with Dirichlettype boundary conditions. Conclusion. A new method of bifurcation analysis for nonlinear partial differential equations has been developed. Specifically, a method has been proposed for reducing partial different equations to ordinary equations, which allows the use of the developed apparatus of bifurcation analysis for boundary value problems of ordinary differential equations. This method allows conducting bifurcation analysis for arbitrary nonlinear partial differential equations.

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

Convergence analysis of an iterative scheme for solving initial value problem for multidimensional partial differential equations

Existence of solutions of initial value problems for nonlinear fractional differential equations on the half-axis

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

There are considered controlled systems with distributed parameters in one dimension. Their mathematical model is given by mixed initial boundary value problems for hyperbolic partial differential equations in two dimensions. The control is performed at the boundaries, being described by ordinary differential equations accounting for the dynamics of the controllers. Two are the standard problems for such systems, especially in the nonlinear case: well posedness in the sense of J. Hadamard (existence,uniqueness and smooth dependence on data) and stability of the steady state solutions - a control oriented property. While feedback control synthesis may be performed at the formal level, validation of the resulting closed loop model as well as stability have to be analyzed with the mathematical rigor required by the theory of partial differential equations. The present paper is focused on these topics starting from the basic model of the flexible beam, accounting for the models of the flexible manipulator arm, of the overhead crane and of the oilwell drillstring. It is shown how various approaches that are known from the fundamentals of partial differential equations can be used to obtain the necessary properties.

Most of the physical processes arising in nature are modeled by either ordinary or partial differential equations. From the point of view of analog computability, the existence of an effective way to obtain solutions of these systems is essential. A pioneering model of analog computation is the General Purpose Analog Computer (GPAC), introduced by Shannon as a model of the Differential Analyzer and improved by Pour-El, Lipshitz and Rubel, Costa and Gra\c{c}a and others. Its power is known to be characterized by the class of differentially algebraic functions, which includes the solutions of initial value problems for ordinary differential equations. We address one of the limitations of this model, concerning the notion of approximability, a desirable property in computation over continuous spaces that is however absent in the GPAC. In particular, the Shannon GPAC cannot be used to generate non-differentially algebraic functions which can be approximately computed in other models of computation. We extend the class of data types using networks with channels which carry information on a general complete metric space $X$; for example $X=C(R,R)$, the class of continuous functions of one real (spatial) variable. We consider the original modules in Shannon's construction (constants, adders, multipliers, integrators) and we add \emph{(continuous or discrete) limit} modules which have one input and one output. We then define an L-GPAC to be a network built with $X$-stream channels and the above-mentioned modules. This leads us to a framework in which the specifications of such analog systems are given by fixed points of certain operators on continuous data streams. We study these analog systems and their associated operators, and show how some classically non-generable functions, such as the gamma function and the zeta function, can be captured with the L-GPAC.

The mathematical modeling of numerous real-world applications results in hierarchical optimization problems with two decision makers where at least one of them has to solve an optimal control problem of ordinary or partial differential equations. Such models are referred to as bilevel optimal control problems. Here, we first review some different features of bilevel optimal control including important applications, existence results, solution approaches, and optimality conditions. Afterwards, we focus on a specific problem class where parameters appearing in the objective functional of an optimal control problem of partial differential equations have to be reconstructed. After verifying the existence of solutions, necessary optimality conditions are derived by exploiting the optimal value function of the underlying parametric optimal control problem in the context of a relaxation approach.

In this paper, we introduce the operators related to mixed boundary value problems for general linear elliptic partial complex differential equations in the unit disc of the complex plane. The solvability of the relevant boundary value problems will be studied by transforming them into singular integral equations.

A numerical method is suggested for solving systems of nonautonomous loaded linear ordinary differential equations with nonseparated multipoint and integral conditions. The method is based on the convolution of integral conditions into local ones. As a result, the original problem is reduced to an initial value (Cauchy) problem for systems of ordinary differential equations and linear algebraic equations. The approach proposed is used in combination with the linearization method to solve systems of loaded nonlinear ordinary differential equations with nonlocal conditions. An example of a loaded parabolic equation with nonlocal initial and boundary conditions is used to show that the approach can be applied to partial differential equations. Numerous numerical experiments on test problems were performed with the use of the numerical formulas and schemes proposed.

Numerical ranges and stability estimates

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.

In this article, we describe on a state of the art of validated numerical computations for solutions of differential equations. A brief overview of the main techniques for self-validating numerics for initial and boundary value problems in ordinary and partial differential equations including eigenvalue problems will be presented. A fairly detailed introductions are given for the author's own method related to second-order elliptic boundary value problems. Many references which seem to be useful for readers are supplied at the end of the article.