Initial Value Problems For ODEs Research Articles

A New Hybrid Block Method for Solving First-Order Differential System Models in Applied Sciences and Engineering

This paper presents a new hybrid block method formulated in variable stepsize mode to solve some first-order initial value problems of ODEs and time-dependent partial differential equations in applied sciences and engineering. The proposed method is implemented considering an adaptive stepsize strategy to maintain the estimated error in each step within a specified tolerance. In order to evaluate the performance and usefulness of the proposed technique in real-world applications, several differential problems from applied sciences and engineering, such as the SIR model, Jacobi elliptic function problem, and chemical reactions problems, are solved numerically. The results of numerical simulations in this work demonstrate that the proposed method is more efficient than other existing numerical methods used for comparisons.

An adaptive optimized Nyström method for second‐order IVPs

This research work deals with the development, analysis, and implementation of an adaptive optimized one‐step Nyström method for solving second‐order initial value problems of ODEs and time‐dependent partial differential equations. The new method is developed through a collocation technique with a new approach for selecting the collocation points. An embedding‐like procedure is used to estimate the error of the proposed optimized method. The current approach has been used to compute efficiently approximate solutions to general second‐order IVPs. The numerical experiments demonstrate that the introduced error estimation and step‐size control strategy presented in this manuscript have produced a good performance compared to some of the other existing numerical methods.

On the solution of Volterra integral equations with decomposable kernel functions

We consider the solution of a class of nonlinear Volterra integral equations, x(t)=g(t)+∫⁡t0tk(t,s;x(s))ds, where t≥t0 and the kernel function k is finitely decomposable, and derive variation of parameters formulae that provide the solution of corresponding perturbed nonlinear equations. We attain this by relating the integral equations to a certain class of initial value problems for ODEs, for which variation of parameters are also formulated.

A tenth order [formula omitted]-stable two-step hybrid block method for solving initial value problems of ODEs

In this article, a new two-step hybrid block method for the numerical integration of ordinary differential initial value systems is presented. The method is obtained after considering two intermediate points and the approximation of the true solution by an adequate polynomial and imposing collocation conditions. The proposed method has the tenth algebraic order of convergence and is A-stable. The numerical experiments considered revealed the superiority of the new method for solving this kind of problems, in comparison with methods of similar characteristics appeared in the literature.

On the General Solution of Initial Value Problems of Ordinary Differential Equations Using the Method of Iterated Integrals

Our goal is to give a very simple, effective and intuitive algorithm for the solution of initial value problem of ODEs of 1st and arbitrary higher order with general i.e. constant, variable or nonlinear coefficients. We find a local expansion of the differential equation/function that employs the higher derivatives (in the sense of total derivatives) of the differential equation/function with respect to independent variable to find an approximation which can be successively improved to desired accuracy by adding extra terms. The method can also be used easily for general 2nd and higher order ODEs. We explain with examples the steps to solution of initial value problems of 1st order ODEs and later follow with more examples for linear and non-linear 2nd order ODEs.

A new stepsize change technique for Adams methods

Abstract In this paper a new technique for stepsize changing in the numerical solution of Initial Value Problems for ODEs by means of Adams type methods is considered. The computational cost of the new technique is equivalent to those of the well known interpolation technique (IT). It is seen that the new technique has better stability properties than the IT and moreover, its leading error term is smaller. These facts imply that the new technique can outperform the IT.

A unified approach for the development of [formula omitted]-step block Falkner-type methods for solving general second-order initial-value problems in ODEs

A family of k-step block multistep methods where the main formulas are of Falkner type is proposed for the direct integration of the general second order initial-value problem where the differential equation is of the general form y″=f(x,y,y′). The two main Falkner formulas and the additional ones to complete the block procedure are obtained from a continuous approximation derived via interpolation and collocation at k+1 points. The main characteristics of the methods are discussed through their formulation in vector form. Each method is formulated as a group of 2k simultaneous formulas over k non-overlapping intervals. In this way, the method produces the approximation of the solution simultaneously at k points on these intervals. As in other block methods, there is no need of other procedures to provide starting approximations, and thus the methods are self-starting (sharing this advantage of Runge–Kutta methods). The resulting family is efficient and competitive compared with other existing methods in the literature, as may be seen from the numerical results.

A Sixth-order Self-Starting Algorithms for Second Order Initial Value Problems of ODEs

A Sixth-order Self-Starting Algorithms for Second Order Initial Value Problems of ODEs

A first approach in solving initial-value problems in ODEs by elliptic fitting methods

Exponentially-fitted and trigonometrically-fitted methods have a long successful history in the solution of initial-value problems, but other functions might be considered in adapted methods. Specifically, this paper aims at the derivation of a new numerical scheme for approximating initial value problems of ordinary differential equations using elliptic functions. The example considered is the undamped Duffing equation where the forcing term is of autonomous type affected by a perturbation parameter. The new scheme is constructed by considering a suitable approximation to the theoretical solution based on elliptic functions. The proposed elliptic fitting procedure has been tested on a variety of problems, showing its good performance.

A Numerical Framework for Integrating Deferred Correction Methods to Solve High Order Collocation Formulations of ODEs

Recent analysis and numerical experiments show that the deferred correction methods are competitive numerical schemes for time dependent differential equations. These methods differ in the mathematical formulations, choices of collocation points, and numerical integration or differentiation strategies. Existing analyses of these methods usually follow traditional ODE theory and study each algorithm's convergence and stability properties as the step size $$\varDelta t$$Δt varies. In this paper, we study the deferred correction methods from a different perspective by separating two different concepts in the algorithm: (1) the properties of the converged solution to the collocation formulation, and (2) the convergence procedure utilizing the deferred correction schemes to iteratively and efficiently reduce the error in the provisional solution. This new viewpoint allows the construction of a numerical framework to integrate existing techniques, by (1) selecting an appropriate collocation discretization based on the physical properties of the solution to balance the time step size and accuracy of the initial approximate solution; and by (2) applying different deferred correction strategies for reducing different components in the error of the provisional solution. This paper discusses properties of different components in the numerical framework, and presents preliminary results on the effective integration of these components for ODE initial value problems. Our results provide useful guidelines for implementing "optimal" time integration schemes for general time dependent differential equations.

Silent error detection in numerical time-stepping schemes

Errors due to hardware or low-level software problems, if detected, can be fixed by various schemes, such as recomputation from a checkpoint. Silent errors are errors in application state that have escaped low-level error detection. At extreme scale, where machines can perform astronomically many operations per second, silent errors threaten the validity of computed results. We propose a new paradigm for detecting silent errors at the application level. Our central idea is to frequently compare computed values to those provided by a cheap checking computation, and to build error detectors based on the difference between the two output sequences. Numerical analysis provides us with usable checking computations for the solution of initial-value problems in ODEs and PDEs, arguably the most common problems in computational science. Here, we provide, optimize, and test methods based on Runge–Kutta and linear multistep methods for ODEs, and on implicit and explicit finite difference schemes for PDEs. We take the heat equation and Navier–Stokes equations as examples. In tests with artificially injected errors, this approach effectively detects almost all meaningful errors, without significant slowdown.

On "Solving differential equations in R" by K.~Soetaert, J.~Cash, and F.~Mazzia

There is a number of powerful software packages intended to solve numerical problems and differential equations in particular. First of all, we can mention here MATLAB, Maple and Mathematica. Recently to that list we can add R. The advantage of using R as compared to the previously mentioned programs, which are rather expensive, is that R is an open source software. At the same time, the recent development of R make it a very powerful and flexible software for solving differential equations. The book ''Solving differential equations in R by K. Soetaert (NIOO-CEME, Yerseke, Netherlands); J. Cash (Imperial College, London, UK); F. Mazzia (University of Bari, Italy), Springer, World Scientific, Publishers Series in Business, 2012. Approx. 250 p. 70 illus. (Use R!) Softcover. ISBN: 978-3-642-28069-6, describes the present stage of development of R packages intended to solve differential equations. Consecutive chapters focus on initial value problems for ODE, differential algebraic equations, boundary value problems for ODE, delayed differential equations and partial differential equations. From the point of view of the R software the book presents the following R packages: \textbf{deSolve}, \textbf{rootSolve}, \textbf{deTestSet}, \textbf{ReacTran} and \textbf{bvpSolve}. The package \textbf{deSolve} contains most of the integration methods used to solve differential equations. It contains a large number of Runge-Kutta methods, both explicit and implicit, and a large variety of multistep methods (Adams methods). These methods can be applied to solve initial value problems for both non-stiff and stiff ODE systems. Also delayed differential equations can be solved using specially designed functions form the package \textbf{deSolve}. Algebraic differential equations can be solved using special methods from \textbf{deSolve} and \textbf{deTestSet} packages. The package \textbf{bvpSolve} is for solving boundary value problems. This package implements 3 methods of solution: shooting, MIRK and collocation methods. Solving partial differential equations in R requires an additional work from the user. The idea is to reduce a particular PDE to a system of ODE's or a system of algebraic equations and then solve such a system using tools available in R. For simple parabolic or hyperbolic equations the method of line can be used and reduce the problem to a system of ODE's which can be solved by special functions from the \textbf{deSolve} package. For elliptic equations a finite difference approximations can be applied and the resulting system of algebraic equations can be solved by a function from the package \textbf{rootSolve}. A grid generation for diffusive and adjective transport terms in analyzed PDE's can be performed automatically by functions from the package \textbf{ReacTran}. Subsequent topics of the book are presented in a similar manner. First comes a chapter with relevant theoretical results about existence of solutions, limits of solvability, discretization methods and stability problems. The presentation is accompanied by a large number of references where more detailed information is available. Then follows a chapter with R implementation of described methods together with a detailed information about the use of the relevant packages. These implementations are illustrated by a large number of well-known examples. Most of important ODE's models are treated as illustrations to particular methods or peculiarities of solutions obtained. Logistic equation, Lorenz model, Arenstorf orbits, van der Pol equation, Josephson junctions, pendulum problem, Sturm-Liouville problem and a large number of models from biology and medicine -- all that can be found in some place in the book. To whom the book is intended? First of all to practitioners: researchers and graduate students who need solutions to differential equations. The R software gives them a possibility to obtain solutions without going into details of computer coding. In addition, a large number of R graphic and statistical functions can improve visualization of the solution obtained. There is however a word of warning to non-expert in numerical analysis. There is a number of traps in numerical solutions of differential equations related to solvability or numerical stability problems. Majority of these traps is addressed in theoretical chapters of the book. But these theoretical chapters are written in a very concise form. A mathematical theorem is sometime reported just in one sentence. Hence, when the reader is a non-expert he should read very carefully a relevant part of theoretical introduction before starting computations to avoid frustration when the result do not fit to expectations. The book can be also very interesting for experts in differential equations. I strongly recommend the book to all courses in ODE and numerical ODE as a valuable source of interesting examples and illustrations to the course. It can be as well an interesting source of problems for students enabling them to test lectured topics and experiment with new models. Finally even experts in the field can find in the book tasty remarks on the state of the art of some numerical methods for differential equations.

Two convergence theorems of variational iteration method for ordinary differential equations

In this article, two convergence theorems of He’s variational iteration method for initial-value problems in ordinary differential equations and first-order ordinary differential equations are presented. It has been shown that the convergence of the variational iteration method only requires the right-hand side of the ODE be Lipschitz continuous for initial-value problems in ODEs.

On the Maxwell−Stefan Approach to Diffusion: A General Resolution in the Transient Regime for One-Dimensional Systems

The diffusion process in a multicomponent system can be formulated in a general form by the generalized Maxwell-Stefan equations. This formulation is able to describe the diffusion process in different systems, such as, for instance, bulk diffusion (in the gas, liquid, and solid phase) and diffusion in microporous materials (membranes, zeolites, nanotubes, etc.). The Maxwell-Stefan equations can be solved analytically (only in special cases) or by numerical approaches. Different numerical strategies have been previously presented, but the number of diffusing species is normally restricted, with only few exceptions, to three in bulk diffusion and to two in microporous systems, unless simplifications of the Maxwell-Stefan equations are considered. In the literature, a large effort has been devoted to the derivation of the analytic expression of the elements of the Fick-like diffusion matrix and therefore to the symbolic inversion of a square matrix with dimensions n x n (n being the number of independent components). This step, which can be easily performed for n = 2 and remains reasonable for n = 3, becomes rapidly very complex in problems with a large number of components. This paper addresses the problem of the numerical resolution of the Maxwell-Stefan equations in the transient regime for a one-dimensional system with a generic number of components, avoiding the definition of the analytic expression of the elements of the Fick-like diffusion matrix. To this aim, two approaches have been implemented in a computational code; the first is the simple finite difference second-order accurate in time Crank-Nicolson scheme for which the full mathematical derivation and the relevant final equations are reported. The second is based on the more accurate backward differentiation formulas, BDF, or Gear's method (Shampine, L. F. ; Gear, C. W. SIAM Rev. 1979, 21, 1.), as implemented in the Livermore solver for ordinary differential equations, LSODE (Hindmarsh, A. C. Serial Fortran Solvers for ODE Initial Value Problems, Technical Report; https://computation.llnl.gov/casc/odepack/odepack_ home.html (2006).). Both methods have been applied to a series of specific problems, such as bulk diffusion of acetone and methanol through stagnant air, uptake of two components on a microporous material in a model system, and permeation across a microporous membrane in model systems, both with the aim to validate the method and to add new information to the comprehension of the peculiar behavior of these systems. The approach is validated by comparison with different published results and with analytic expressions for the steady-state concentration profiles or fluxes in particular systems. The possibility to treat a generic number of components (the limitation being essentially the computational power) is also tested, and results are reported on the permeation of a five component mixture through a membrane in a model system. It is worth noticing that the algorithm here reported can be applied also to the Fick formulation of the diffusion problem with concentration-dependent diffusion coefficients.

An application of Taylor models to the Nakao method on ODEs

The authors give short survey on validated computation of initial value problems for ODEs especially Taylor model methods. Then they propose an application of Taylor models to the Nakao method which has been developed for numerical verification methods on PDEs and apply it to initial value problems for ODEs with some numerical experiments.

A “Discretization” Technique for the Solution of ODEs II

A functional analytic technique was recently presented for finding discrete equivalent counterparts of initial value problems of ODEs and obtaining their real analytic solutions. In the current paper, this technique is extended to boundary value problems of ODEs and to the complex solutions of ODEs. In order to demonstrate this technique, it is applied to the classic Blasius problem of fluid mechanics. Apart from its real solution, its complex solution is also studied. The obtained results indicate that the complex Blasius function exhibits an oscillatory behavior and strengthen a conjecture regarding its singularities in the complex plane.

Optimal adaptive solution of initial-value problems with unknown singularities

The optimal solution of initial-value problems in ODEs is well studied for smooth right-hand side functions. Much less is known about the optimality of algorithms for singular problems. In this paper, we study the (worst case) solution of scalar problems with a right-hand side function having r continuous bounded derivatives in R , except for an unknown singular point. We establish the minimal worst case error for such problems (which depends on r similarly as in the smooth case), and define optimal adaptive algorithms. The crucial point is locating an unknown singularity of the solution by properly adapting the grid. We also study lower bounds on the error of an algorithm for classes of singular problems. In the case of a single singularity with nonadaptive information, or in the case of two or more singularities, the error of any algorithm is shown to be independent of r.

Validated solutions of initial value problems for parametric ODEs

In initial value problems for ODEs with interval-valued parameters and/or initial values, it is desirable in many applications to be able to determine a validated enclosure of all possible solutions to the ODE system. Much work has been done for the case in which initial values are given by intervals, and there are available software packages that deal with this case. However, less work has been done on the case in which parameters are given by intervals. We describe here a new method for obtaining validated solutions of initial value problems for ODEs with interval-valued parameters. The method also accounts for interval-valued initial values. The effectiveness of the method is demonstrated using several numerical examples involving parametric uncertainties.

Energy conservative algorithm for numerical solution of ODEs initial value problems

The numerical treatment of the ODE initial value problems is an intensively researched field. Recently the qualitative algorithms, such as monotonicity and positivity preserving algorithms are in the focus of investigation. For the dynamical systems the energy conservative algorithms are very important. In the case of Hamiltonian system the symplectic algorithms are very effective. This kind of algorithm is not adaptive, but doubtless they are powerful. The high-efficiency computers and the computer algebraic software systems allow us to create efficient adaptive energy conservative numerical algorithm for solving ODE initial value problems. In this article an adaptive numerical-analytical algorithm is suggested which very effectively can be applied for Hamiltonian systems, but the idea of construction can be adaptable for other initial value problems too, where there are some quantity preserved in time. The idea and the efficiency of the proposed algorithm will be presented by simple examples, such as the Lotka-Volterra and linear oscillator problems.

Blended implicit methods for the numerical solution of DAE problems

Recently, a new approach for solving the discrete problems, generated by the application of block implicit methods for the numerical solution of initial value problems for ODEs, has been devised [L. Brugnano, Blended block BVMs (B 3 VMs): a family of economical implicit methods for ODEs, J. Comput. Appl. Math. 116 (2000) 41–62; L. Brugnano, C. Magherini, Blended implementation of block implicit methods for ODEs, Appl. Numer. Math. 42 (2002) 29–45; L. Brugnano, D. Trigiante, Block implicit methods for ODEs, in: D. Trigiante (Ed.), Recent Trends in Numerical Analysis, Nova Science Publishers, New York, 2001, pp. 81–105]. This approach is based on the so-called blended implementation of the methods, giving corresponding blended implicit methods. The latter have been implemented in the computational code BiM [L. Brugnano, C. Magherini, The BiM code for the numerical solution of ODEs, J. Comput. Appl. Math. 164–165 (2004) 145–158]. Blended implicit methods are here extended to handle the numerical solution of DAE problems, resulting in a straightforward generalization of the basic approach.