Order Order Ordinary Differential Equations Research Articles

Analysis of Forchheimer Effect for Double Diffusive Convection With Dusty Fluids and MHD

An attempt has been made to analyze the effect of second order resistance for a steady, dusty fluid considering magnetohydrodynamic (MHD) and also the characteristics of fluid like permeability, porosity, solutal diffusivity and thermal conductivity being varied. Here the basic equations are coupled, non-linear Partial Differential Equations (PDEs), which are changed by similarity transformations to higher order Ordinary Differential Equations (ODE). After being transformed the higher order ODE that where obtained are resolved numerically. Shooting technique is employed here and the values are tabulated for various pertinent parameter variations. The effects of the inertia, concentration and interaction, mixed convection, magnetic and many other parameters are discussed and plotted graphs for velocity, concentration and temperature. The magnetic force enhances velocity. It was found that the present study correlates with the existence results.

The effect of MHD mixed convection flow over a heated plate

AbstractThe notion of this paper is to analyse the bidimensional, steady flow of a dusty fluid over a heated plate with varied fluid properties and magnetic fields. The boundary layer flow directed by Lapwood–Forcheimer–Brinkman extended Darcy model and Lorentz force are highly coupled and nonlinear. Similarity terms convert these nonlinear partial differential equations to higher‐order ordinary differential equations. These higher order ordinary differential equations (ODEs) are converted to first order ODE by the numerical technique called the shooting technique. Temperature, concentration, and velocity are portrayed for various pertinent parameters, and it is spotted that due to the particles in the fluid, the heat transfer speeds up and temperature and concentration decrease. Nusselt number, Skin fiction, and Sherwood number also have been discussed here. The results are in keeping with the existing results. In understanding the thermal characteristic of various effects, it was seen that thermogeometric plate, radiation conduction is very useful for many industrial applications.

An analysis of errors for pre-service teachers in first order ordinary differential equations

Literature has established that some learners encountered difficulties solving first order ordinary differential equations (ODEs). The use of error analysis in teaching ODEs is believed to make essential contribution towards calculus knowledge development. This paper therefore focuses on analyzing pre-service teachers’ (PSTs) errors and misconceptions apropos of first order ODEs. The paper analyzed the nature of errors made in a test which was written by PSTs on the above topic. The test comprised various types of first order differential equations such as ODEs with separable variables, exact ODEs, ODEs that needed integrating factors, linear ODEs, and homogeneous ODEs. The purpose was to investigate the challenges faced by PSTs in various types of ODEs and the nature of misconceptions that they had in each particular type. This is a qualitative study that involved 63 PSTs who wrote a test on ODEs after being taught the topic for two weeks. The authors marked the work in order to ascertain the misconceptions and errors exhibited by the participants in the test. The PSTs’ performance in the test was analyzed using the SOLO taxonomy and the Newman’s theory mistake analysis. The study established that the topic was rather difficult for PSTs due to various reasons that included, among others, knowledge gaps in integration rules, algebraic computations and, in rare cases, differentiation, as well as misapplication of the rules of natural logarithms. This research therefore recommends that mathematics teacher educators ought to rather focus on the concept of integration and basic algebra before introducing the topic on ODEs to teachers on training.

Decoupling of second order systems via linear time invariant transformations

Decoupling of linear time invariant (LTI) systems of second order ordinary differential equations (ODEs) with real matrix coefficients via LTI transformations in the configuration space is revisited. Necessary and sufficient conditions for the existence of such transformations are presented, as well as methods to construct transformation matrices. Matrix series and general matrix function formulas for the system’s matrix coefficients which enable decoupling are also discussed. Systems with non-symmetric and symmetric matrices are considered. Situations when systems degenerate into second order ODEs and lower order ODEs, including differential algebraic equations (DAEs), are analyzed as well.

Approximating non linear higher order ODEs by a three point block algorithm.

Differential equations are commonly used to model various types of real life applications. The complexity of these models may often hinder the ability to acquire an analytical solution. To overcome this drawback, numerical methods were introduced to approximate the solutions. Initially when developing a numerical algorithm, researchers focused on the key aspect which is accuracy of the method. As numerical methods becomes more and more robust, accuracy alone is not sufficient hence begins the pursuit of efficiency which warrants the need for reducing computational cost. The current research proposes a numerical algorithm for solving initial value higher order ordinary differential equations (ODEs). The proposed algorithm is derived as a three point block multistep method, developed in an Adams type formulae (3PBCS) and will be used to solve various types of ODEs and systems of ODEs. Type of ODEs that are selected varies from linear to nonlinear, artificial and real life problems. Results will illustrate the accuracy and efficiency of the proposed three point block method. Order, stability and convergence of the method are also presented in the study.

Comparison of Noether Symmetries and First Integrals of Two-Dimensional Systems of Second Order Ordinary Differential Equations by Real and Complex Methods

Noether symmetries and first integrals of a class of two-dimensional systems of second order ordinary differential equations (ODEs) are investigated using real and complex methods. We show that first integrals of systems of two second order ODEs derived by the complex Noether approach cannot be obtained by the real methods. Furthermore, it is proved that a complex method can be extended to larger systems and higher order.

Implicit Two-point Block Method with Third and Fourth Derivatives for Solving General Second Order ODEs

In this paper, we present an implicit two-point block method for solving directly the general second order ordinary differential equations (ODEs). The method incorporates the first and second derivatives of f(x; y; y'), which are the third and fourth derivatives of the solution. The method is derived using Hermite Interpolating Polynomial as the basic function. A performance comparison of the two-point block method is compared in term of accuracy to several existing methods, which have order almost equal or higher than that of the new method. Numerical results interpret the accuracy and efficacy of the new method. Application of the new method is discussed.

Block Hybrid Method with Trigonometric-Fitting for Solving Oscillatory Problems

In this paper, we develop algebraic order conditions for two-point block hybrid method up to order five using the approach of B-series. Based on the order conditions, we derive fifth order two-point block explicit hybrid method for solving special second order ordinary differential equations (ODEs), where the existing explicit hybrid method of order five is used to be the method at the first point. The method is then trigonometrically fitted so that it can be suitable for solving highly oscillatory problems arising from special second order ODEs. The new trigonometrically-fitted block method is tested using a set of oscillatory problems over a very large interval. Numerical results clearly showed the superiority of the method in terms of accuracy and execution time compared to other existing methods in the scientific literature.

Kantorovich variational method for the flexural analysis of CSCS Kirchhoff-Love plates

The Kantorovich variational method was used in this study to solve the flexural problem of Kirchhoff-Love plates with two opposite edges x=±a/2 clamped and the other two edges y=±b/2 simply supported, for the case of uniformly distributed transverse load over the entire plate domain. The plate considered was assumed homogeneous, and isotropic. The total potential energy functional for the Kirchhoff-Love plate was found as the sum of the potential energy of the applied distributed load and the strain energy functional of the plate. The symmetrical nature of the plate and the load was used to choose the deflection function as a single series of infinite terms involving the cosine function of the y coordinate variable with the unknown function being dependent on the x coordinate variable. Integration with respect to the y coordinate variable simplified the total potential energy functional to be dependent on the unknown function of x, and its derivatives. Euler-Lagrange differential equations were used to find the differential equations of equilibrium of the plate as a system of homogeneous fourth order ordinary differential equations (ODE) in the unknown function. The system of fourth order ODE was solved using the method of differential D operators, or the method of trial functions to obtain the general solution as the linear superposition of the homogeneous and particular solutions. The demands of symmetry of unknown function about the point x= 0, and the boundary conditions at the clamped edges were used to obtain all the four integration constants; thus, completely determining the deflection. Bending moment distributions were determined using the bending moment deflection relations. The deflection was determined at the centre of the plate. The bending moment values were also determined at the centre of the plate, and at the middle of the clamped edges. Expressions obtained for the deflection at the centre, and bending moment values were found to be single series of infinite terms with highly converging properties. The solutions obtained converged to the exact solutions with the use of only four terms of the series.

A New Integrator for Special Third Order Differential Equations With Application to Thin Film Flow Problem

In recent time, Runge-Kutta methods that integrate special third order ordinary differential equations (ODEs) directly are proposed to address efficiency issues associated with classical Runge-Kutta methods. Albeit, the methods require evaluation of three set of equations to proceed with the numerical integration. In this paper, we propose a class of multistep-like Runge-Kutta methods (hybrid methods), which integrates special third order ODEs directly. The method is completely derivative-free. Algebraic order conditions of the method are derived. Using the order conditions, a four-stage method is presented. Numerical experiment is conducted on some test problems. The method is also applied to a practical problem in Physics and engineering to ascertain its validity. Results from the experiment show that the new method is more accurate and efficient than the classical Runge-Kutta methods and a class of direct Runge-Kutta methods recently designed for special third order ODEs.

Hybrid methods for direct integration of special third order ordinary differential equations

In this paper we present a new class of direct numerical integrators of hybrid type for special third order ordinary differential equations (ODEs), y′′′=f(x,y); namely, hybrid methods for solving third order ODEs directly (HMTD). Using the theory of B-series, order of convergence of the HMTD methods is investigated. The main result of the paper is a theorem that generates algebraic order conditions of the methods that are analogous to those of two-step hybrid method. A three-stage explicit HMTD is constructed. Results from numerical experiment suggest the superiority of the new method over several existing methods considered in the paper.

Classification of scalar third order ordinary differential equations linearizable via generalized contact transformations

Whereas Lie had linearized scalar second order ordinary differential equations (ODEs) by point transformations, and later Chern had extended to the third order by using contact transformation, till recently no work had been done for higher order (or systems) of ODEs. Lie had found a unique class defined by the number of infinitesimal symmetry generators but the more general ODEs were not so classified. Recently, classifications of higher order and systems of ODEs were provided. In this paper we relate contact symmetries of scalar ODEs with point symmetries of reduced systems. We define a new type of transformation that builds upon this relation and obtain equivalence classes of scalar third order ODEs linearizable via these transformations. Four equivalence classes of such equations are seen to exist.

Numerical Algorithm of Block Method for General Second Order ODEs using Variable Step Size

This paper outlines an alternative algorithm for solving general second order ordinary differential equations (ODEs). Normally, the numerical method was designed for solving higher order ODEs by converting it into an n-dimensional first order equations with implementation of constant step length. Nevertheless, this involved a lot of computational complexity which led to consumption a lot of time. Consequently, a direct block multistep method with utilization of variable step size strategy is proposed. This method was developed for computing the solution at four points simultaneously and the derivation based on numerical integration as well as using interpolation approach. The convergence of the proposed method is justified under suitable conditions of stability and consistency. Five numerical examples are considered and some comparisons are made with the existing methods for demonstrating the validity and reliability of the proposed algorithm.

NUMERICAL METHOD FOR SINGULARLY PERTURBED THIRD ORDER ORDINARY DIFFERENTIAL EQUATIONS OF REACTION-DIFFUSION TYPE

In this paper, we have proposed a numerical method for Singularly Perturbed Boundary Value Problems (SPBVPs) of reaction-diffusion type of third order Ordinary Differential Equations (ODEs). The SPBVP is reduced into a weakly coupled system of one first order and one second order ODEs, one without the parameter and the other with the parameter <TEX>${\varepsilon}$</TEX> multiplying the highest derivative subject to suitable initial and boundary conditions, respectively. The numerical method combines boundary value technique, asymptotic expansion approximation, shooting method and finite difference scheme. The weakly coupled system is decoupled by replacing one of the unknowns by its zero-order asymptotic expansion. Finally the present numerical method is applied to the decoupled system. In order to get a numerical solution for the derivative of the solution, the domain is divided into three regions namely two inner regions and one outer region. The Shooting method is applied to two inner regions whereas for the outer region, standard finite difference (FD) scheme is applied. Necessary error estimates are derived for the method. Computational efficiency and accuracy are verified through numerical examples. The method is easy to implement and suitable for parallel computing. The main advantage of this method is that due to decoupling the system, the computation time is very much reduced.

Weak formulation and first order form of the equations of motion for a class of constrained mechanical systems

Some new theoretical results are presented on modeling the dynamic response of a class of discrete mechanical systems subject to equality motion constraints. Both the development and presentation are facilitated by employing some fundamental concepts of differential geometry. At the beginning, the equations of motion of the corresponding unconstrained system are presented on a configuration manifold with general properties, first in strong and then in a primal weak form, using Newton׳s law of motion as a foundation. Next, the final weak form is obtained by performing a crucial integration by parts step, involving a covariant derivative. This step required the clarification and enhancement of some concepts related to the variations employed in generating the weak form. The second part of this work is devoted to systems involving holonomic and non-holonomic scleronomic constraints. The equations of motion derived in a recent study of the authors are utilized as a basis. The novel characteristic of these equations is that they form a set of second order ordinary differential equations (ODEs) in both the coordinates and the Lagrange multipliers associated to the constraint action. Based on these equations, the corresponding weak form is first obtained, leading eventually to a consistent first order ODE form of the equations of motion. These equations are found to appear in a form resembling the form obtained after application of the classical Hamilton׳s canonical equations. Finally, the new theoretical findings are illustrated by three representative examples.

Linearization of two dimensional complex-linearizable systems of second order ordinary differential equations

Complex-linearization of a class of systems of second order ordinary differential equations (ODEs) has already been studied with complex symmetry analysis. Linearization of this class has been achieved earlier by complex method, however, linearization criteria and the most general linearizable form of such systems have not been derived yet. In this paper, it is shown that the general linearizable form of the complex-linearizable systems of two second order ODEs is (at most) quadratically semi-linear in the first order derivatives of the dependent variables. Further, linearization conditions are derived in terms of coefficients of system and their derivatives. These linearizable 2-dimensional complex-linearizable systems of second order ODEs are characterized here, by adopting both the real and complex procedures.

Embedded explicit Runge–Kutta type methods for directly solving special third order differential equations [formula omitted

In this paper three pairs of embedded Runge–Kutta type methods for directly solving special third order ordinary differential equations (ODEs) of the form y‴=f(x,y) denoted as RKD methods are presented. The first is the RKD4(3) pair which is third order embedded in fourth-order method has the property first same as last (FSAL) whereby the last row of the coefficient matrix is equal to the vector output. The second method is the RKD5(4) pair followed by the RKD6(5) pair. The methods are derived with the strategies such that the higher order methods are very accurate and the lower order methods will give the best error estimates. Variables stepsize codes are developed based on the methods and used to solve a set of special third order problems. Numerical results are compared with the existing embedded Runge–Kutta pairs which require the problems to be reduced into a system of first order ODEs. Numerical results have clearly shown the advantage and the efficiency of the new RKD pairs.

A general framework for the numerical solution of second order ODEs

In this paper the authors consider the family of general linear methods (GLMs) for special second order ordinary differential equations (ODEs) of the type y′′=f(y(t)), recently introduced with the aim to provide an unifying formulation for numerical methods solving such problems and achieve a general strategy for the analysis of the minimal demandings in terms of accuracy and stability to be asked for, such as consistency, zero-stability and convergence. They emphasize the generality of this approach, by showing that the family of GLMs for second order ODEs recovers classical numerical formulae known in the literature and allows to easily obtain new methods by proving their convergence in a simple, straightforward way.

Order conditions for General Linear Nyström methods

The purpose of this paper is to analyze the algebraic theory of order for the family of general linear Nystrom (GLN) methods introduced in D'Ambrosio et al. (Numer. Algorithm 61(2), 331–349, 2012) with the aim to provide a general framework for the representation and analysis of numerical methods solving initial value problems based on second order ordinary differential equations (ODEs). Our investigation is carried out by suitably extending the theory of B-series for second order ODEs to the case of GLN methods, which leads to a general set of order conditions. This allows to recover the order conditions of numerical methods already known in the literature, but also to assess a general approach to study the order conditions of new methods, simply regarding them as GLN methods: the obtained results are indeed applied to both known and new methods for second order ODEs.

Solving Nonstiff Higher Order Odes Using Variable Order Step Size Backward Difference Directly

The current numerical techniques for solving a system of higher order ordinary differential equations (ODEs) directly calculate the integration coefficients at every step. Here, we propose a method to solve higher order ODEs directly by calculating the integration coefficients only once at the beginning of the integration and if required once more at the end. The formulae will be derived in terms of backward difference in a constant step size formulation. The method developed will be validated by solving some higher order ODEs directly using variable order step size. To simplify the evaluations of the integration coefficients, we find the relationship between various orders. The results presented confirmed our hypothesis.