Michael Chapwanya, Robert Dozva and Gift Muchatibaya
This paper aims to design new finite difference schemes for the Lane–Emden type equations. In particular, the authors show that the schemes are stable with respect to the…
Abstract
Purpose
This paper aims to design new finite difference schemes for the Lane–Emden type equations. In particular, the authors show that the schemes are stable with respect to the properties of the equation. The authors prove the uniqueness of the schemes and provide numerical simulations to support the findings.
Design/methodology/approach
The Lane–Emden equation is a well-known highly nonlinear ordinary differential equation in mathematical physics. Exact solutions are known for a few parameter ranges and it is important that any approximation captures the properties of the equation it represent. For this reason, designing schemes requires a careful consideration of these properties. The authors apply the well-known nonstandard finite difference methods.
Findings
Several interesting results are provided in this work. The authors list these as follows. Two new schemes are designed. Mathematical proofs are provided to show the existence and uniqueness of the solution of the discrete schemes. The authors show that the proposed method can be extended to singularly perturbed equations.
Originality/value
The value of this work can be measured as follows. It is the first time such schemes have been designed for the kind of equations.
Details
Keywords
A.A. Aderogba, M. Chapwanya and J.K. Djoko
For a partial differential equation with a fourth-order derivative such as the Cahn-Hilliard equation, it is always a challenge to design numerical schemes that can handle the…
Abstract
Purpose
For a partial differential equation with a fourth-order derivative such as the Cahn-Hilliard equation, it is always a challenge to design numerical schemes that can handle the restrictive time step introduced by this higher order term. The purpose of this paper is to employ a fractional splitting method to isolate the convective, the nonlinear second-order and the fourth-order differential terms.
Design/methodology/approach
The full equation is then solved by consistent schemes for each differential term independently. In addition to validating the second-order accuracy, the authors will demonstrate the efficiency of the proposed method by validating the dissipation of the Ginzberg-Lindau energy and the coarsening properties of the solution.
Findings
The scheme is second-order accuracy, the authors will demonstrate the efficiency of the proposed method by validating the dissipation of the Ginzberg-Lindau energy and the coarsening properties of the solution.
Originality/value
The authors believe that this is the first time the equation is handled numerically using the fractional step method. Apart from the fact that the fractional step method substantially reduces computational time, it has the advantage of simplifying a complex process efficiently. This method permits the treatment of each segment of the original equation separately and piece them together, in a way that will be explained shortly, without destroying the properties of the equation.
Details
Keywords
Tadeusz Sobczyk and Marcin Jaraczewski
Discrete differential operators (DDOs) of periodic functions have been examined to solve boundary-value problems. This paper aims to identify the difficulties of using those…
Abstract
Purpose
Discrete differential operators (DDOs) of periodic functions have been examined to solve boundary-value problems. This paper aims to identify the difficulties of using those operators to solve ordinary nonlinear differential equations.
Design/methodology/approach
The DDOs have been applied to create the finite-difference equations and two approaches have been proposed to reduce the Gibbs effects, which arises in solutions at discontinuities on the boundaries, by adding the buffers at boundaries and applying the method of images.
Findings
An alternative method has been proposed to create finite-difference equations and an effective method to solve the boundary-value problems.
Research limitations/implications
The proposed approach can be classified as an extension of the finite-difference method based on the new formulas approximating the derivatives. This can be extended to the 2D or 3D cases with more flexible meshes.
Practical implications
Based on this publication, a unified methodology for directly solving nonlinear partial differential equations can be established.
Originality/value
New finite-difference expressions for the first- and second-order derivatives have been applied.
Details
Keywords
Marcin Jaraczewski and Tadeusz Sobczyk
Discrete differential operators of periodic base functions have been examined to solve boundary-value problems. This paper aims to identify the difficulties of using those…
Abstract
Purpose
Discrete differential operators of periodic base functions have been examined to solve boundary-value problems. This paper aims to identify the difficulties of using those operators to solve ordinary linear and nonlinear differential equations with Dirichlet and Neumann boundary conditions.
Design/methodology/approach
This paper presents a promising approach for solving two-dimensional (2D) boundary problems of elliptic differential equations. To create finite differential equations, specially developed discrete partial differential operators are used to replace the partial derivatives in the differential equations. These operators relate the value of the partial derivatives at each point to the value of the function at all points evenly distributed over the area where the solution is being sought. Exemplary 2D elliptic equations are solved for two types of boundary conditions: the Dirichlet and the Neumann.
Findings
An alternative method has been proposed to create finite-difference equations and an effective method to determine the leakage flux in the transformer window.
Research limitations/implications
The proposed approach can be classified as an extension of the finite-difference method based on the new formulas approximating the derivatives. This method can be extended to the 3D or time-periodic 2D cases.
Practical implications
This paper presents a methodology for calculations of the self- and mutual-leakage inductances for windings arbitrarily located in the transformer window, which is needed for special transformers or in any case of the internal asymmetry of windings.
Originality/value
The presented methodology allows us to obtain the magnetic vector potential distribution in the transformer window only, for example, to omit the magnetic core of the transformer from calculations.