Second-order Initial Value Problems Research Articles

A Parametric Six-Step Method for Second-Order IVPs with Oscillating Solutions

In this paper, we develop an explicit symmetric six-step method for the numerical solution of second-order initial value problems (IVPs) with oscillating solutions. The proposed method is phase-fitted and incorporates a free coefficient as a parameter to optimize its performance. By exploring a wide range of values for this parameter, we computationally determine the periodicity interval. The objective of this procedure is to identify the range of the parameter’s values for which the method remains stable. Based on the output from the periodicity interval analysis, we then aim to define the optimal values for the parameter by numerically solving three initial value problems. The results guided us in identifying these optimal values and confirm the high efficiency of the new method. The method’s efficiency is further validated for the chosen optimal parameter value for specific oscillatory problems, where it is compared with well-known phase-fitted methods.

An Effective Matrix Technique for the Numerical Solution of Second Order Differential Equations

In this paper, an effective technique for solving differential equations with initial conditions is presented. The method is based on the use of the Legendre matrix of derivatives defined on the close interval [-1,1]. Properties of the polynomial are outlined and further used to obtain the matrix of derivative which was used in transforming the differential equation into systems of linear and nonlinear algebraic equations. The systems of these algebraic equations were then solved using Gaussian elimination method to determine the unknown parameters required for approximating the solution of the differential equation. The advantage of this technique over other methods is that, it has less computational manipulations and complexities and also its availability for application on both linear and nonlinear second-order initial value problems is impressive. Other advantage of the algorithm is that high accurate approximate solutions are achieved by using a greater number of terms of the Legendre polynomial and once the operational matrix is obtained, it can be used to solve differential equations of higher order by introducing just a little manipulation on the operational matrix. Some existing sample problems from literature were solved and the results were compared to show the validity, simplicity and applicability of the proposed method. The results obtained validate the simplicity and applicability of the method and it also reveals that the method perform better than most existing methods.

Hp-version C1-continuous Petrov–Galerkin method for nonlinear second-order initial value problems with application to wave equations

<i>hp</i>-version <i>C</i>1-continuous Petrov–Galerkin method for nonlinear second-order initial value problems with application to wave equations

Postprocessing Techniques of High-Order Galerkin Approximations to Nonlinear Second-Order Initial Value Problems with Applications to Wave Equations

Postprocessing Techniques of High-Order Galerkin Approximations to Nonlinear Second-Order Initial Value Problems with Applications to Wave Equations

One-step block scheme with optimal hybrid points for numerical integration of second-order ordinary differential equations

In this paper, a one-step block of optimized hybrid schemes for the numerical integration of second-order initial value problems (IVP) of ordinary differential equations (ODE) is constructed via collocation techniques. The developed scheme is obtained by considering two intra-step nodal points as hybrid points, which are chosen in order to achieve optimized errors of the main formulae approximating the solution such that 0 &lt; v1 &lt; v 2 &lt; 1 where v1 and v2 are defined as hybrid points. The characteristics of the developed scheme are analyzed. Application of the new scheme on some second-order IVPs shows the accuracy and effectiveness of the scheme compared with some existing methods.

Residual-based a posteriori error analysis of an ultra-weak discontinuous Galerkin method for nonlinear second-order initial-value problems

Residual-based a posteriori error analysis of an ultra-weak discontinuous Galerkin method for nonlinear second-order initial-value problems

Directly Self-Starting Second-Order Explicit Integration Methods with Dissipation Control and Adjustable Bifurcation Points for Second-Order Initial Value Problems

This paper proposes a directly self-starting second-order [Formula: see text]-sub-step explicit method, within which the three novel members are developed and analyzed for demonstration. The novel explicit schemes can output second-order accelerations for general structures, which cannot be achieved for the directly self-starting single- and two-sub-step algorithms. Each explicit algorithm achieves controllable algorithmic dissipation and adjustable bifurcation points. As a result, the novel explicit algorithms impose two algorithmic parameters, [Formula: see text] and [Formula: see text], to control numerical dissipation and bifurcation point, respectively. The parameter [Formula: see text], denoting the spectral radius at the bifurcation point, controls numerical dissipation, whereas the parameter [Formula: see text], denoting the bifurcation point, adjusts the amount of dissipation imposed in the low-frequency range. This paper provides users with two recommended selections of [Formula: see text]. Apart from these desirable features, each novel explicit algorithm attains the maximum stability bound, [Formula: see text] where [Formula: see text] denotes the number of sub-steps. The four- and five-sub-step algorithms perform optimization to reduce numerical low-frequency dissipation. As the number of sub-steps increases, the novel explicit algorithms can have better numerical characteristics without an increase in the computational effort. Numerical examples are simulated to validate the performance and superiority of the novel explicit algorithms.

Numerical methods for solving second-order initial value problems of ordinary differential equations with Euler and Runge-Kutta fourth-order methods

This paper presents two standard numerical methods for solving second order initial value problems for ordinary differential equations (ODEs). The Euler and the Runge-Kutta fourth-order methods are applied without any discretization or restrictive assumptions for solving ODEs. The numerical solutions obtained by the two methods are in good agreement with the exact solutions. The convergence and error analysis which are discussed demonstrate the effectiveness of the methods. The results obtained from the two numerical methods show that the RK4 method is appropriate, consistent, convergent, quite stable, and more accurate than the Euler's method.

A promising exponentially-fitted two-derivative Runge–Kutta–Nyström method for solving [formula omitted]: Application to Verhulst logistic growth model

Explicit exponentially-fitted two-derivative Runge–Kutta–Nyström method with single f-function and multiple third derivatives is proposed for solving special type of second-order ordinary differential equations with exponential solutions. B-series and rooted tree theory for the proposed method are developed for the derivation of order conditions. Then, we build frequency-dependent coefficients for the proposed method by integrating the second-order initial value problem exactly with solution in the linear composition of set functions eλt and e−λt with λ∈R. An exponentially-fitted two-derivative Runge–Kutta–Nyström method with three stages fifth order is derived. Linear stability and stability region of the proposed method are analyzed. The numerical tests show that the proposed method is more effective than other existing methods with similar algebraic order in the integration of special type of second-order ordinary differential equations with exponential solutions. Also, the proposed method is used to solve a famous application problem, Verhulst logistic growth model and the result shows the proposed method still works effectively for solving this model.

Reproducing kernel-based piecewise methods for efficiently solving oscillatory systems of second-order initial value problems

Reproducing kernel-based piecewise methods for efficiently solving oscillatory systems of second-order initial value problems

A One-Point Third-Derivative Hybrid Multistep Technique for Solving Second-Order Oscillatory and Periodic Problems

This paper describes a third-derivative hybrid multistep technique (TDHMT) for solving second-order initial-value problems (IVPs) with oscillatory and periodic problems in ordinary differential equations (ODEs), the coefficients of which are independent of the frequency omega and step size h . This research is significant because it has numerous applications to real-life phenomena such as chaotic dynamical systems, almost periodic problems, and duffing equations. The current method is derived from the collocation of a derivative function at the equidistant grid and off-grid points. The TDHMT obtained is a continuous scheme for obtaining simultaneous approximations to the solution and its derivative at each point in the x 0 , x N interval integration. The presence of high derivatives increases the order of the method, which increases the accuracy method’s order and the stability property, as discussed in detail. Finally, the proposed method is compared to existing methods in the literature on some oscillatory and periodic test problems to demonstrate the technique’s effectiveness and productivity.

Odd Order Integrator with Two Complex Functions Control Parameters for Solving Systems of Initial Value Problems

In this study, a numerical integrator that is based on a nonlinear interpolant, for the local representation of the theoretical solution is presented. The resulting integrator aims to solve second and higher-order initial value problems as systems of first-order initial value problems. The method is designed to have two complex functions as control parameters. The control parameters may become real, depending on the nature of the second-order initial value problems to be solved. The generalization and properties of the scheme are also presented.

A Single-Step Modified Block Hybrid Method for General Second-Order Ordinary Differential Equations

A multistep collocation approach is used to derive a single-step modified block hybrid method (MBHM) of order five for solving general second-order initial-value problems (IVPs) of ordinary differential equations (ODEs). The new method's basic convergence property is established, and its numerical accuracy is demonstrated using numerical examples from the literature. The new method outperforms similar methods in terms of accuracy, earning it a recommendation as a likely candidate for solving general second-order ODEs.

Fifth Order Improved Runge-Kutta Nystrom Method Using Trigonometrically-Fitting for Solving Oscillatory Problems

In this paper, the Trigonometrically Fitted Improved Runge-KuttaNystrom method is proposed as a novel method with four stages and fifth order for solving oscillatory problems. This method is intended to integrate second-order initial value problems using the trigonometrically fitting approach. To increase the method'saccuracy, the principal frequency of the problem푤∈ℝ, is used. It is discovered that the new method is more precise when compared with the other existing Runge-Kutta Nystrom and IRKN5 methods. To show how well the TFIRKN5 method works, test problems for second-order ordinary differential equations (ODEs) are solved. The numerical outcomes show that the novel approach outperforms methods that have already been published.

Adaptive multi-step Runge–Kutta–Nyström methods for general second-order ordinary differential equations

In this study, a variable step size formulation of multi-step general Runge–Kutta–Nyström (MSGN) methods to directly integrate general second-order initial value problems (IVPs) is considered. This formula is carried out using an embedded explicit pair where: the higher-order formula is an accurate and the lower-order formula uses to estimate the local error. Numerical results show that the new methods perform much better in terms of functions evaluations, implementation cost and execution time compared to other existing high quality embedded Runge–Kutta (ERK) methods in the literature.

Superconvergence of an Ultra-Weak Discontinuous Galerkin Method for Nonlinear Second-Order Initial-Value Problems

In this paper, we develop and analyze an ultra-weak discontinuous Galerkin (UWDG) method for nonlinear second-order initial-value problems for ordinary differential equations of the form [Formula: see text]. Our main concern is to study the convergence and superconvergence properties of the proposed scheme. With a suitable choice of the numerical fluxes, we prove the optimal error estimates with order [Formula: see text] in the [Formula: see text]-norm for the solution, when piecewise polynomials of degree at most [Formula: see text] are used. We use these results to prove that the UWDG solution is superconvergent with order [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text] towards a special projection of the exact solution. We further prove that the [Formula: see text]-degree UWDG solution and its derivative are [Formula: see text] superconvergent at the end of each step. Our proofs are valid for arbitrary regular meshes using piecewise polynomials with degree [Formula: see text]. Finally, numerical experiments are provided to verify that all theoretical findings are sharp. The main advantage of our method over the standard DG method for systems of first-order equations is that the UWDG method can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system, which reduces memory and computational costs.

On a type-2 fuzzy approach to solution of second-order initial value problem

On a type-2 fuzzy approach to solution of second-order initial value problem

Algorithms of algebraic order nine for numerically solving second-order boundary and initial value problems in ordinary differential equations

Algorithms of algebraic order nine for numerically solving second-order boundary and initial value problems in ordinary differential equations

A Functionally-Fitted Block Numerov Method for Solving Second-Order Initial-Value Problems with Oscillatory Solutions

A functionally-fitted Numerov-type method is developed for the numerical solution of second-order initial-value problems with oscillatory solutions. The basis functions are considered among trigonometric and hyperbolic ones. The characteristics of the method are studied, particularly, it is shown that it has a third order of convergence for the general second-order ordinary differential equation, y''=f left( x,y,y' right) , it is a fourth order convergent method for the special second-order ordinary differential equation, y''=f left( x,yright) . Comparison with other methods in the literature, even of higher order, shows the good performance of the proposed method.

A shooting approach to the scaled boundary finite element equations of elastodynamics in the frequency domain

Dealing with numerical analysis of problems, especially ones with semi-infinite boundaries, scaled boundary finite element method has emerged as one of the efficient tools for the task. Combining the exactness of strong forms with the flexibility of weak formulations makes the method an improvement to its predecessors. Problem with the method arises when the analytical solution of the semi-discretized system is not available, which is the case for numerous problems. In the most recent attempt to solve the issue, a shooting method was proposed for elastostatic problems. Generality of the method removes any concerns regarding the type of governing equations since it no longer needs any analytical solutions. In this study, the method is further developed to deal with elastodynamic problems of viscoelastic media in the frequency domain, for which the closed-form solution is not available. Adopting the shooting method, the two-point boundary value problem of each subdomain is treated as a second-order initial value problem. Boundary conditions of the bounded subdomains are formulated by enforcing equilibrium at both ends. On the other hand, in contrast to linear elasticity, neither forces nor displacements vanish at infinity. As a result, a special treatment is required to deal with the boundary conditions of the unbounded subdomains. A Runge–Kutta–Nyström method, which consists of an embedded pair of adjacent order, is utilized to integrate the strong form of oscillatory equations enabling local error estimation and adaptive step-size control. This guarantees efficiency since the integration proceeds not by equal step-sizes, but by equal user-defined local errors. In the first step, analytical solution of a viscoelastic half-space subjected to a harmonic strip load is used for the verification process. First, the half-space is discretized by Voronoi cells and each truncated edge is defined using an unbounded subdomain. Next, one bounded subdomain is used to define the whole bounded area, and one unbounded subdomain resembles the radiation condition of the far-field boundaries. For the cases involving only bounded subdomains, three other numerical tests, including a perforated plate, an L-shaped panel, and a bi-material specimen are employed, and the effect of discretization method on the results is analyzed. The method is a huge step forward since the size of subdomains are not restricted, far-field boundaries can be arbitrary-shaped, hysteretic damping can be naturally included, no additional unknowns are introduced, no compound matrices are defined, no preconditioning matrices are needed, and a simple unified approach is used for both bounded and unbounded subdomains.