Geometric Integration

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

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 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.

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

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.

Singular perturbed problems in ordinary differential equations

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

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

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.

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.

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 search for efficient higher order methods is a constant goal in numerical analysis. In this paper, a higher order two‐step hybrid block method is presented to directly solve second‐order initial value problems in ordinary differential equations. In addition to the higher order, the proposed method has been formulated in variable step‐size mode to extract its best performance. Comparisons with other methods in the literature show the good accuracy it can provide. Theoretical aspects such as linear stability and convergence analysis are also discussed.

Dynamic properties of the local linearization method for initial-value problems

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