Pedogogical perspectives on the method of Wilmer III and Costa for solving second-order differential equations

Recently, Wilmer III and Costa introduced a method into the mathematics education research literature which they employed to construct solutions to certain classes of ordinary differential equations. In this article, we build on their ideas in the following ways. We establish a link between their approach and the method of successive approximations. We show how applying the method of approximations naturally leads to the constructed approximation of Wilmer III and Costa. The new link is important because it enables us to respond to several challenges posed by Wilmer III and Costa. This includes addressing issues raised therein with convergence of their recursively constructed sequence of functions, and responding to their call regarding more mathematical rigour when relaxing the polynomial condition on the coefficients in the differential equation. Furthermore, the new link is pedagogically significant because it also opens up new pedagogical points of view. For example, the results in this paper provide potentially alternate, but overlapping, perspectives that are suitable for, and jointly inform, the learning and teaching of solution methods to differential equations. The value of this is supported by the presumption of Tisdell that teaching multiple ways to solve the same problem has academic and social value.     

A comparison between the variational iteration method and the successive approximations method

In this paper we investigate and compare the variational iteration method and the successive approximations method for solving a class of nonlinear differential equations. We prove that these two methods are equivalent for solving these types of equations.     

New uniqueness results for fractional differential equations

We develop the Krasnoselskii–Krein type of uniqueness theorem for an initial value problem of the Riemann–Liouville type fractional differential equation which involves a function of the form f?(t,?x(t),?D ^q?1 x(t)), for 1<q<2 and establish the convergence of successive approximations. We prove a few other uniqueness theorems.     

An efficient step method for a system of differential equations with delay

Using the step method, we study a system of delay differential equations and we prove the existence and uniqueness of the solution and the convergence of the successive approximation sequence using the Perov''s contraction principle and the step method. Also, we propose a new algorithm of successive approximation sequence generated by the step method and, as an example, we consider some second order delay differential equations with initial conditions.     

Newton iteration for partial differential equations and the approximation of the identity

It is known that the critical condition which guarantees quadratic convergence of approximate Newton methods is an approximation of the identity condition. This requires that the composition of the numerical inversion of the Fréchet derivative with the derivative itself approximate the identity to an accuracy calibrated by the residual. For example, the celebrated quadratic convergence theorem of Kantorovich can be proven when this holds, subject to regularity and stability of the derivative map. In this paper, we study what happens when this condition is not evident a priori but is observed a posteriori. Through an in-depth example involving a semilinear elliptic boundary value problem, and some general theory, we study the condition in the context of dual norms, and the effect upon convergence. We also discuss the connection to Nash iteration.     

Comparison between variational iteration method and homotopy perturbation method for linear and nonlinear partial differential equations with the nonhomogeneous initial conditions

In this study, linear and nonlinear partial differential equations with the nonhomogeneous initial conditions are considered. We used Variational iteration method (VIM) and Homotopy perturbation method (HPM) for solving these equations. Both methods are used to obtain analytic solutions for different types of differential equations. Four examples are presented to show the application of the present techniques. In these schemes, the solution takes the form of a convergent series with easily computable components. The present methods perform extremely well in terms of efficiency and simplicity. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

An accessible,justifiable and transferable pedagogical approach for the differential equations of simple harmonic motion

Recently, Gauthier introduced a method to construct solutions to the equations of motion associated with oscillating systems into the mathematics education research literature. In particular, Gauthier's approach involved certain manipulations of the differential equations; and drew on the theory of complex variables.

Motivated by the work of Gauthier, we construct an alternative pedagogical approach for the learning and teaching of solution methods to these equations. The innovation lies in drawing on factorization techniques of differential equations and harmonizing them with Gauthier's approach of the theory of complex variables. When blended together to form a new approach, the significance lies in its accessibility, justifiability and transferrability to other problems.

We pedagogically ground our approach in the educational development theory of Piaget, with the results informing the learning and teaching of solution methods to differential equations for lecturers, teachers and learners within universities, colleges, polytechnics and schools around the world.     

Fitted Galerkin spectral method to solve delay partial differential equations

In this paper, we consider a class of parabolic partial differential equations with a time delay. The first model equation is the mixed problems for scalar generalized diffusion equation with a delay, whereas the second model equation is a delayed reaction‐diffusion equation. Both of these models have inherent complex nature because of which their analytical solutions are hardly obtainable, and therefore, one has to seek numerical treatments for their approximate solutions. To this end, we develop a fitted Galerkin spectral method for solving this problem. We derive optimal error estimates based on weak formulations for the fully discrete problems. Some numerical experiments are also provided at the end. Copyright © 2015 John Wiley & Sons, Ltd.     

A numerical method to solve a class of linear integro‐differential equations with weakly singular kernel

In this paper, a collocation method based on the Bessel polynomials is introduced for the approximate solution of a class of linear integro‐differential equations with weakly singular kernel under the mixed conditions. The exact solution can be obtained if the exact solution is polynomial. In other cases, increasing number of nodes, a good approximation can be obtained with applicable errors. In addition, the method is presented with error and stability analysis. Copyright © 2012 John Wiley & Sons, Ltd.     

A weak condition for secant method to solve systems of nonlinear equations

In this paper, a new weak condition for the convergence of secant method to solve the systems of nonlinear equations is proposed. A convergence ball with the center x0 is replaced by that with xl, the first approximation generated by the secant method with the initial data x-1 and x0. Under the bounded conditions of the divided difference, a convergence theorem is obtained and two examples to illustrate the weakness of convergence conditions are provided. Moreover, the secant method is applied to a system of nonlinear equations to demonstrate the viability and effectiveness of the results in the paper.     

A collocation method to solve higher order linear complex differential equations in rectangular domains

In this article, a collocation method is developed to find an approximate solution of higher order linear complex differential equations with variable coefficients in rectangular domains. This method is essentially based on the matrix representations of the truncated Taylor series of the expressions in equation and their derivates, which consist of collocation points defined in the given domain. Some numerical examples with initial and boundary conditions are given to show the properties of the method. All results were computed using a program written in scientific WorkPlace v5.5 and Maple v12. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

A discrete and q asymptotic iteration method

ABSTRACT

We introduce a finite difference and q-difference analogues of the Asymptotic Iteration Method of Ciftci, Hall, and Saad. We give necessary, and sufficient condition for the existence of a polynomial solution to a general linear second-order difference or q-difference equation subject to a ‘terminating condition’, which is precisely defined. When a difference or q-difference equation has a polynomial solution, we show how to find the second solution.     

Perron type theorems for functional differential equations with infinite delay in a Banach space

For functional differential equations in a Banach space, we obtain some results on the asymptotic behavior of solutions, which correspond to Perron type theorems for differential equations without delay or with finite delay.     

Jacobi rational approximation and spectral method for differential equations of degenerate type

We introduce an orthogonal system on the half line, induced by Jacobi polynomials. Some results on the Jacobi rational approximation are established, which play important roles in designing and analyzing the Jacobi rational spectral method for various differential equations, with the coefficients degenerating at certain points and growing up at infinity. The Jacobi rational spectral method is proposed for a model problem appearing frequently in finance. Its convergence is proved. Numerical results demonstrate the efficiency of this new approach.

     

Application of generalized differential transform method to multi-order fractional differential equations

In a recent paper [Odibat Z, Momani S, Erturk VS. Generalized differential transform method: application to differential equations of fractional order, Appl Math Comput. submitted for publication] the authors presented a new generalization of the differential transform method that would extended the application of the method to differential equations of fractional order. In this paper, an application of the new technique is applied to solve fractional differential equations of the form y^(μ)(t)=f(t,y(t),y^(β₁)(t),y^(β₂)(t),…,y^(β_n)(t)) with μ>β_n>β_n-1>…>β₁>0, combined with suitable initial conditions. The fractional derivatives are understood in the Caputo sense. The method provides the solution in the form of a rapidly convergent series. Numerical examples are used to illustrate the preciseness and effectiveness of the new generalization.     

Application to differential transformation method for solving systems of differential equations

In this paper, we present an analytical solution for different systems of differential equations by using the differential transformation method. The convergence of this method has been discussed with some examples which are presented to show the ability of the method for linear and non-linear systems of differential equations. We begin by showing how the differential transformation method applies to a non-linear system of differential equations and give two examples to illustrate the sufficiency of the method for linear and non-linear stiff systems of differential equations. The results obtained are in good agreement with the exact solution and Runge–Kutta method. These results show that the technique introduced here is accurate and easy to apply.     

Tight estimates for general averaging applied to almost-periodic differential equations

Estimates are presented for the averaging method of ordinary differential equations. Previous results are improved by relaxing the conditions under which they hold, and by providing tight bounds for the estimations for almost-periodic differential equations and quasi-periodic differential equations.     

On application of the Lanczos method to solution of some partial differential equations

Let A be a square symmetric n × n matrix, φ be a vector from ⁿ, and f be a function defined on the spectral interval of A. The problem of computation of the vector u = f(A)φ arises very often in mathematical physics.

We propose the following method to compute u. First, perform m steps of the Lanczos method with A and φ. Define the spectral Lanczos decomposition method (SLDM) solution as u_m = φ Qf(H)e₁, where Q is the n × m matrix of the m Lanczos vectors and H is the m × m tridiagonal symmetric matrix of the Lanczos method. We obtain estimates for u − u_m that are stable in the presence of computer round-off errors when using the simple Lanczos method.

We concentrate on computation of exp(− tA)φ, when A is nonnegative definite. Error estimates for this special case show superconvergence of the SLDM solution. Sample computational results are given for the two-dimensional equation of heat conduction. These results show that computational costs are reduced by a factor between 3 and 90 compared to the most efficient explicit time-stepping schemes. Finally, we consider application of SLDM to hyperbolic and elliptic equations.