Local error estimates for moderately smooth problems: Part I – ODEs and DAEs

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.

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.

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.

List of Examples Preface 1. Introduction. Boundary Value Problems for Ordinary Differential Equations Boundary Value Problems in Applications 2. Review of Numerical Analysis and Mathematical Background. Errors in Computation Numerical Linear Algebra Nonlinear Equations Polynomial Interpolation Piecewise Polynomials, or Splines Numerical Quadrature Initial Value Ordinary Differential Equations Differential Operators and Their Discretizations 3. Theory of Ordinary Differential Equations. Existence and Uniqueness Results Green's Functions Stability of Initial Value Problems Conditioning of Boundary Value Problems 4. Initial Value Methods. Introduction. Shooting Superposition and Reduced Superposition Multiple Shooting for Linear Problems Marching Techniques for Multiple Shooting The Riccati Method Nonlinear Problems 5. Finite Difference Methods. Introduction Consistency, Stability, and Convergence Higher-Order One-Step Schemes Collocation Theory Acceleration Techniques Higher-Order ODEs Finite Element Methods 6. Decoupling. Decomposition of Vectors Decoupling of the ODE Decoupling of One-Step Recursions Practical Aspects of Consistency Closure and Its Implications 7. Solving Linear Equations. General Staircase Matrices and Condensation Algorithms for the Separated BC Case Stability for Block Methods Decomposition in the Nonseparated BC Case Solution in More General Cases 8. Solving Nonlinear Equations. Improving the Local Convergence of Newton's Method Reducing the Cost of the Newton Iteration Finding a Good Initial Guess Further Remarks on Discrete Nonlinear BVPS 9. Mesh Selection. Introduction Direct Methods A Mesh Strategy for Collocation Transformation Methods General Considerations 10. Singular Perturbations. Analytical Approaches Numerical Approaches Difference Methods Initial Value Methods 11. Special Topics. Reformulation of Problems in 'Standard' Form Generalized ODEs and Differential Algebraic Equations Eigenvalue Problems BVPs with Singularities Infinite Intervals Path Following, Singular Points and Bifurcation Highly Oscillatory Solutions Functional Differential Equations Method of Lines for PDEs Multipoint Problems On Code Design and Comparison Appendix A. A Multiple Shooting Code Appendix B. A Collocation Code References Bibliography Index.

Reachability analysis is a fundamental problem for safety verification and falsification of Cyber-Physical Systems (CPS) whose dynamics follow physical laws usually represented as differential equations. In the last two decades, numerous reachability analysis methods and tools have been proposed for a common class of dynamics in CPS known as ordinary differential equations (ODE). However, there is lack of methods dealing with differential algebraic equations (DAE), which is a more general class of dynamics that is widely used to describe a variety of problems from engineering and science, such as multibody mechanics, electrical circuit design, incompressible fluids, molecular dynamics, and chemical process control. Reachability analysis for DAE systems is more complex than ODE systems, especially for high-index DAEs because they contain both a differential part (i.e., ODE) and algebraic constraints (AC). In this paper, we propose a scalable reachability analysis for a class of high-index large linear DAEs. In our approach, a high-index linear DAE is first decoupled into one ODE and one or several AC subsystems based on the well-known Marz decoupling method utilizing admissible projectors. Then, the discrete reachable set of the DAE, represented as a list of star-sets, is computed using simulation. Unlike ODE reachability analysis where the initial condition is freely defined by a user, in DAE cases, the consistency of the initial condition is an essential requirement to guarantee a feasible solution. Therefore, a thorough check for the consistency is invoked before computing the discrete reachable set. Our approach successfully verifies (or falsifies) a wide range of practical, high-index linear DAE systems in which the number of state variables varies from several to thousands.

In recent years, powerful interval arithmetic tools have been developed for the computation of guaranteed enclosures of the sets of all reachable states of dynamical systems. In such simulations, uncertainties in initial conditions and parameters are taken into account by worst-case bounds. The resulting enclosures are verified in the sense that all reachable states are guaranteed to be included. This is achieved by taking into account both the influence of the above-mentioned uncertainties as well as numerical inaccuracies arising from computer implementations using finite-precision floating-point arithmetic. In this contribution, a computational framework for both offline and online applications of interval tools in control design is presented. Verified computational procedures and their applications to the solution of initial value problems for both ordinary differential equations and differential algebraic equations are summarized. These algorithms are employed for verified feedforward control design as well as state and disturbance estimation for a distributed heating system.KeywordsInterval ArithmeticRobust ControllerFeedforward ControlReachable StateInterval MethodThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The HBT(10)9 method for ODEs is expanded into HBT(10)9DAE for solving nonstiff and moderately stiff systems of fully implicit differential algebraic equations (DAEs) of arbitrarily high fixed index. A scheme to generate first-order derivatives at off-step points is combined with Pryce scheme which generates high order derivatives at step points. The stepsize is controlled by a local error estimator. HBT(10)9DAE uses only the first four derivatives of y instead of the first 10 required by Taylor’s series method T10DAE of order 10. Dormand–Prince’s DP(8,7)13M for ODEs is extended to DP(8,7)DAE for DAEs. HBT(10)9DAE wins over DP(8,7)DAE on several test problems on the basis of CPU time as a function of relative error at the end of the interval of integration. An index-5 problem is equally well solved by HBT(10)9DAE and T10DAE. On this problem, the error in the solution by DP(8,7)DAE increases as time increases.

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 one-step algorithm for solving ordinary differential equations with initial conditions is developed. Given the p first derivatives of y(x), the unknown function determined by the differential equation, this algorithm provides a method with a local discretization error of order 2p+1 instead of p+1 as is to be expected if we use the standard Taylor method with p derivatives. Several examples are given which illustrate the behaviour of the proposed algorithm and are also used to compare the efficiency of this algorithm, in terms of the computer time needed to attain a given final error tolerance, with other commonly applied techniques, such as Runge-Kutta or Adams-Moulton. The range of applicability of the method is also considered.

In recent papers the technique for a local and global error estimation and the local-global step size control were presented to solve both ordinary differential equations and semi-explicit index 1 differential-algebraic systems by multistep methods with any reasonable accuracy attained automatically. Now those results are extended to the concept of multistep extrapolation, and the paper demonstrates with numerical examples how such methods work in practice. Especially, we develop an efficient technique to calculate higher derivatives of a numerical solution with Hermite interpolating polynomials. The necessary theory is also provided.

Computational methods for reaction–diffusion problems for fourth order ordinary differential equations with a small parameter at the highest derivative

Classical considerations of stability in ODE initial and boundary problems are mirrored by corresponding properties (stiff stability, di- stability) in problem discretizations. However, computational cate- gories are not precise, and qualitative descriptors such as of moderate size cannot be avoided where size varies with the sensitivity of the Newton iteration in nonlinear problems, for example. Sensitive New- ton iterations require close enough initial estimates. The main tool for providing this in boundary value problems is continuation with respect to a parameter. If stable discretizations are not available, then adaptive meshing is needed to follow rapidly changing solutions. Use of such tools can be necessary in stable nonlinear situations not covered by classical considerations. Implications for the estimation problem are sketched. It is shown how to choose appropriate bound- ary conditions for the embedding method. The simultaneous method is formulated as a constrained optimization problem. It avoids explicit ODE characterization and appears distinctly promising. However, its properties are not yet completely understood.