Inverse Hybrid Linear Multistep Methods for Solving the Second Order Initial Value Problems in Ordinary Differential Equations

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.

We present a software package, M odes , offering h -adaptive and p -adaptive linear multistep methods for first order initial value problems in ordinary differential equations. The implementation is based on a new parametric, grid-independent representation of multistep methods [Arévalo and Söderlind 2017]. Parameters are supplied for over 60 methods. For nonstiff problems, all maximal order methods ( p = k for explicit and p = k +1 for implicit methods) are supported. For stiff computation, implicit methods of order p = k are included. A collection of step-size controllers based on digital filters is provided, generating smooth step-size sequences offering improved computational stability. Controllers may be selected to match method and problem classes. A new system for automatic order control is also provided for designated families of multistep methods, offering simultaneous h - and p -adaptivity. Implemented as a M atlab toolbox, the software covers high order computations with linear multistep methods within a unified, generic framework. Computational experiments show that the new software is competitive and offers qualitative improvements. M odes is available for downloading and is primarily intended as a platform for developing a new generation of state-of-the-art multistep solvers, as well as for true ceteris paribus evaluation of algorithmic components. This also enables method comparisons within a single implementation environment.

A boundary value method (BVM) leads to a discrete boundary value problem with some conditions imposed at the initial points and the remaining ones at the final points. The interest in this class of methods is the fact that BVMs overcome the limitations of the well-known Dahlquist order and stability barrier for an A-stable linear multistep method (LMM). Yet, A-stability is fundamental to the integration of stiff ordinary differential equations (ODEs). This presents multiple families of linear multistep formulas (LMFs) with future points based on extended backward differentiation formula (BDF) which can be used as a main method in a BVM implementation for the solution of initial value problems (IVPs) in ODEs.

This chapter deals with the numerical solution of initial value problems of ordinary differential equations. First, different approaches to obtain integration methods for ordinary differential equations are presented. Then the important properties consistency, stability, and convergence of integration methods are introduced and studied. This allows for a fruitful application of the results obtained in Chapter 12. Subsequently, one-step methods are considered in detail and their important properties are shown. Moreover, Runge–Kutta methods are thoroughly investigated. Finally, we deal with linear multistep methods and study the asymptotic behavior of integration methods and of stiff differential equations. In particular, we deal with Dahlquist’s root condition and derive the first Dahlquist barrier and the second Dahlquist barrier.

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.

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.

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

In this paper, linear multi-step hybrid block methods with three-, four- and five-step numbers are developed for approximating directly the solution of second order Initial and Boundary Value Problems (IBVPs). Multiple finite difference formulas are derived and combined in a block formulation to form a numerical integrator that provides direct solution to second order IBVPs over sub-intervals. A new class of orthogonal polynomials constructed as basis function to develop the hybrid block methods adopting collocation technique with a non-negative weight function. The scheme is applied as simultaneous integrator to second order initial value and boundary value problems of ODEs. The properties and convergence of the proposed method are discussed. The derived schemes were used to solve some problems and the numerical result shows the effectiveness, accuracy and superiority of the method over the existing methods found in the literature.

This paper presented a linear multistep method for solving fourth order initial value problems of ordinary differential equations. Collocation and interpolation methods are adopted in the derivation of the new numerical scheme which is further applied to finding direct solution of fourth order ordinary differentiation equations. This implementation strategy is more accurate and efficient than Adams–Bashforth Method solution. The newly derive scheme have better stabilities properties than that of the Adams-Bashforth Method. Numerical examples are included to illustrate the reliability and accuracy of the new methods.Keywords: Linear Multistep Methods, Region of Absolute Stability, Zero-Stability, Error Analysis, Collocation.

As already mentioned in Chapter 1, we refer to mathematical models that can be expressed as systems of ordinary differential equations and deal with the solution of problems in mechanics and, in general, in applied sciences. Therefore, it is important to deal with mathematical aspects related to the analysis of differential equations and with the solution of initial and boundary-value problems. This chapter refers to the above topics and is organized in seven sections: Section 2.2 deals with a formal classification of ordinary differential equations. Linear and nonlinear properties are identified together with the analysis of some additional properties of the system. Physical interpretations are often related to this type of classification. Section 2.3 deals with the mathematical statement of problems, which is obtained by linking the evolution equation to the conditions that are necessary to its solution. If these conditions are given at the initial value of the independent variable, say t = 0, one has an initial-value problem, also called a Cauchy problem. Otherwise, if these quantities are given at both ends of the range of the independent variable, say t = 0 and t = T, one has a boundary-value problem. The solution of problems has to satisfy both the differential equation and the initial and/or the boundary conditions. This section is mainly devoted to the analytic treatment of the Cauchy problem and will report the basic theorems which provide the conditions necessary to ensure existence, uniqueness, and regularity of the solution. Section 2.4 deals with solution methods of the initial-value problem. Analytic methods can be developed in order to solve linear problems, while computational schemes are needed for nonlinear problems. Actually, this matter is proposed as a concise guide to mathematical methods and to the related literature. The aim of this guide is to provide the necessary elements for the interaction with the scientific computation based on Mathematica ® Section 2.5 deals with the representation of solutions and applications both in the linear and in the nonlinear case. Here, a systematic use of scientific computation is developed. As we shall see, the computation is useful not only in the nonlinear case, but also for the fast treatment of linear problems. Section 2.6 deals with boundary-value problems for ordinary differential equations and the problems related to their solution, both analytical and numerical. The last section suggests some problems which are proposed to the reader’s attention for practicing with mathematical methods.

B-theory of general linear methods for Volterra functional differential equations

This paper constructs highly accurate and efficient time integration methods for the solution of transient problems. The motion equations of transient problems can be described by the first-order ordinary differential equations, in which the right-hand side is decomposed into two parts, a linear part and a nonlinear part. In the proposed methods of different orders, the responses of the linear part at the previous step are transferred by the generalized Padé approximations, and the nonlinear part’s responses of the previous step are approximated by the Gauss–Legendre quadrature together with the explicit Runge–Kutta method, where the explicit Runge–Kutta method is used to calculate function values at quadrature points. For reducing computations and rounding errors, the 2m algorithm and the method of storing an incremental matrix are employed in the calculation of the generalized Padé approximations. The proposed methods can achieve higher-order accuracy, unconditional stability, flexible dissipation, and zero-order overshoots. For linear transient problems, the accuracy of the proposed methods can reach 10−16 (computer precision), and they enjoy advantages both in accuracy and efficiency compared with some well-known explicit Runge–Kutta methods, linear multi-step methods, and composite methods in solving nonlinear problems.

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.

Both Runge-Kutta and linear multistep methods are available to solve initial value problems for ordinary differential equations in the R packages deSolve and deTestSet. Nearly all of these solvers use adaptive step size control, some also control the order of the formula adaptively, or switch between different types of methods, depending on the local properties of the equations to be solved. We show how to trigger the various methods using a variety of applications pointing, where necessary, to problems that may arise. For instance, many practical applications involve discontinuities. As the integration routines assume that a solution is sufficiently differentiable over a time step, handing such discontinuities requires special consideration. We give examples of how we can implement a nonsmooth forcing term, switching behavior, and problems that include sudden jumps in the dependent variables. Since much computational efficiency can be gained by using the correct method for a particular problem, we end this chapter by providing a few guidelines as to how the most efficient solution method for a particular problem can be found.