Additional Separated-Variable Solutions of the Biharmonic Equation in Polar Coordinates

Computational methods for reaction–diffusion problems for fourth order ordinary differential equations with a small parameter at the highest derivative

In this paper, the Elzaki transform method is used for solving two-dimensional (2D) elasticity problems in plane polar coordinates. Airy stress function was used to express the stress compatibility equation as a biharmonic equation. Elzaki transform was applied with respect to the radial coordinate to a modified form of the stress compatibility equation, and the biharmonic equation simplified to a fourth order ordinary differential equation (ODE). The general solution for the Airy stress potential function in the Elzaki transform space was obtained by solving the ODE. By inversion, the general solution for the Airy stress potential function was obtained in the physical domain space variables in terms of four unknown integration constants. Normal stresses and shear stress fields were also determined for the general case of 2D elasticity problems. The Flamant problem was solved as a particular illustration of 2D elasticity problems. The stress boundary conditions and the requirement of equilibrium of the internal stress resultants and the external forces were used simultaneously to determine the four constants of integration. The Airy stress potential function and the normal and shear stress fields were thus completely determined. The principle of superposition was used to obtain the elasticity solutions for the stress fields in the elastic half plane due to strip load of infinite extent, and solutions for horizontal stresses on smooth rigid retaining walls due to strip loads and parallel line load of infinite extent acting on the elastic half plane.

This paper presented a linear multistep method for solving fourth order initial value problems of ordinary differential equations. Collocation and interpolation methods are adopted in the derivation of the new numerical scheme which is further applied to finding direct solution of fourth order ordinary differentiation equations. This implementation strategy is more accurate and efficient than Adams–Bashforth Method solution. The newly derive scheme have better stabilities properties than that of the Adams-Bashforth Method. Numerical examples are included to illustrate the reliability and accuracy of the new methods.Keywords: Linear Multistep Methods, Region of Absolute Stability, Zero-Stability, Error Analysis, Collocation.

The study is devoted to establishing optimal mathematical models in the context of system analysis problems. Namely, nontrivial boundary conditions are applied to the problem of integrating polyharmonic equations in polar coordinates. The function, which is triharmonic in a unit disk, is presented in the form of an integral with a delta-like kernel. The question of the existence of a structural connection between solutions to the triharmonic equations in polar coordinates and positive operators, which are solutions to other partial differential equations, is considered. It is shown that the triharmonic Poisson integral for a unit disk can be represented as an average value of the solution to the Laplace equation in polar coordinates. Keywords: Fourier series, triharmonic equation, unit disk, triharmonic Poisson integral, Abel–Poisson operator.

A new numerical method for solving fourth order ordinary differential equations directly is proposed in this paper. Interpolation and collocation were employed in developing this method using seven steps. The use of the approximated power series as an interpolation equation was adopted in deriving the method. The basic properties of the new method such as zero-stability, consistency, convergence and order are established. The numerical results indicate that the new method gives better accuracy than the existing methods when it is applied to fourth ordinary differential equations.

This paper carries a different approach of collection and interpolation to develop a tenth block method for the numerical solution of linear or nonlinear ordinary differential equations of fourth order with initial conditions. The method has been implemented at the selected mesh points to generate a direct tenth block method through Taylor series. Some critical properties of this method such as zero stability, order of the method, and convergence have been analysed. Two numerical tests have taken to make a comparison of the approximate results with exact as well as results of other authors.

A compact difference scheme for a two dimensional nonlinear fractional Klein–Gordon equation in polar coordinates

The method for range scan segmentation described in a previous paper titled “Improving area center robot navigation using a novel range scan segmentation method”, by some authors of present paper in 2011, uses an extended Kalman filter (EKF) for estimating straight segments equations in polar coordinates. That method, equivalent to a non-linear weighted least squares method, is extended in present work for estimating circle segments equations in polar coordinates, but only to the first stages of the full method, including acquisition of initial estimation of circles centers and radii to be used as initial values for EKF an the sequential iterative estimation of circle equations using EKF. The new problem of variance matrices being almost ill-conditioned for circle equations is addressed with a Tikhonov regularization, or ridge regression. Also a correction of initial values for estimations is proposed to solve the added problems found in circle equations estimation.

Using nonpolynomial cubic spline approximation in space and finite difference in time direction, we discuss three-level implicit difference scheme of O(k2+h4) for the numerical solution of 1D wave equations in polar coordinates, where k>0 and h>0 are grid sizes in time and space coordinates, respectively. The proposed method is applicable to problems with singularity. Stability theory of the proposed method is discussed, and numerical examples are given in support of the theoretical results.

Criteria for fourth-order ordinary differential equations to be linearizable by contact transformations

The block hybrid collocation method with three off-step points is proposed for the direct solution of fourth order ordinary differential equations. The interpolation and collocation techniques are applied on basic polynomial to generate the main and additional methods. These methods are implemented in block form to obtain the approximation at seven points simultaneously. Numerical experiments are conducted to illustrate the efficiency of the method. The method is also applied to solve the fourth order problem from ship dynamics.

Chapter 9 - Differential Equations: Introduction

Fourth-order approximations at first time level, linear stability analysis and the numerical solution of multidimensional second-order nonlinear hyperbolic equations in polar coordinates

Converted state equation Kalman filter for nonlinear maneuvering target tracking

In this paper we present fifth order Runge-Kutta method (RK5) for solving initial value problems of fourth order ordinary differential equations. In this study RK5 method is quite efficient and practically well suited for solving boundary value problems. All mathematical calculation performed by MATLAB software for better accuracy and result. The result obtained, from numerical examples, shows that this method more efficient and accurate. These methods are preferable to some existing methods because of their simplicity, accuracy and less computational cost involved.