We report a new finite-difference approximation of O(h 4 ) for two-dimensional nonlinear triharmonic partial differential equations on a nine-point compact stencil where the values of u, ∂ 2 u/∂n 2 and ∂ 4 u/∂n 4 are prescribed on the boundary. In this method, there is no need to discretize the derivative boundary conditions. The Laplacian and the biharmonic of the solution are obtained as a by-product of the method. We require only a system of three equations to obtain the solution. We compare the advantages and implementation of the proposed method with the corresponding central difference approximations of O(h 2 ) in the context of iterative methods. Numerical results are given to verify the fourth-order convergence rate of the method.
In this article, we present two new novel finite difference approximations of order two and four, respectively, for the three dimensional non-linear triharmonic partial differential equations on a compact stencil where the values of u, ∂2u/∂n2 and ∂4u/∂n4 are prescribed on the boundary. We introduce new ideas to handle the boundary conditions and there is no need to discretize the derivative boundary conditions. We require only 7- and 19-grid points on the compact cell for the second and fourth order approximation, respectively. The Laplacian and the biharmonic of the solution are obtained as by-product of the methods. We require only system of three equations to obtain the solution. Numerical results are provided to illustrate the usefulness of the proposed methods.
This paper deals with a new higher order compact difference scheme, which is, O(h 4 ) using coupled approach on the 19-point 3D stencil for the solution of three dimensional nonlinear biharmonic equations. At each internal grid point, the solution u(x, y, z) and its Laplacian 2 u are obtained. The resulting stencil algorithm is presented and hence this new algorithm can be easily incorporated to solve many problems. The present discretization allows us to use the Dirichlet boundary conditions only and there is no need to discretize the derivative boundary conditions near the boundary. We also show that special treatment is required to handle the boundary conditions. Convergence analysis for a model problem is briefly discussed. The method is tested on three problems and compares very favourably with the corresponding second order approximation which we also discuss using coupled approach.
A white Tiger "Rakesh" aged 13 years and weighing approximately 150kg at Nandankanan Zoo developed lameness in its right hind limb. Oral analgesic (Tramadol hydrochloride 500mg) and antibiotic (Ampoxin 500mg) in beef was administered for five days with no improvement. It was decided to examine the limb physically and radiologically. A mixture of 0.65mg atropine sulphate, 150mg xylazine hydrochloride and 500mg ketamine hydrochloride was darted to the tiger using a pressure gun. After 10 minutes the animal was lifted on a tarpaulin strap to a transportation crate and shifted to the zoo hospital. Both the hind limbs were examined thoroughly for presence of wounds. Radiograph of both the limbs in lateral and dorsal-plantar views were taken starting from tarsal joints to digits. Radiographs revealed a simple incomplete fracture of right third metatarsal bone at its proximal third region. Ringers' Lactate 500ml and DNS 5% 500ml was administered intravenously. The heart rate, respiration rate and temperature varied within normal range during anaesthetic period. Antagozil 1.5ml (15mg) was injected i/v to revive the tiger from anaesthesia after shifting it in a squeeze cage. Cephotaxime 2g, Neurobion 2ml, and Tramadol 2ml (200mg) were injected intramuscularly for seven days. The tiger was kept in squeeze cage with limited movement for one month. Five powdered oral calcium tablets (shellcal-50) in beef was given during the period. The tiger recovered from lameness completely and was released to its enclosure. In the present case since the fracture was incomplete and had no external wound, conservative management was thought to be proper. Dingwell (1974) suggested close reduction, if one or two metatarsal bones are fractured in dogs, with remaining bones acting as support. He further suggested coaptation splint or plaster cast for additional support. As external splints are rarely kept intact by wild animals, confinement of tiger inside a squeeze cage restricted its movement which helped in healing of the incomplete fracture.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.