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FULL LENGTH RESEARCH ARTICLEThe block methods together with their hybrid forms were studied (Onumanyi, et al. 2001) and further investigated Dauda, et al. 2005) and found to be useful for the direct solution of initial and boundary value problems.The advantages of these methods include (i) overcoming the issue of overlap of pieces of solutions usually associated with the multistep finite difference methods and (ii) they are self starting thus eliminating the use of other methods to obtain starting solutions. Before now, the determination of the order of these block methods, their convergence and plotting their absolute stability regions have been done for the single members of the block and whose result may not be assumed for the entire block. In this paper, we consider some of the properties of the entire block.
Convergence of the block linear multistep method:In the discreet case, a linear multistep method is said to be convergent if for all initial value problem (1.0)
ON SOME PROPERTIES OF THE BLOCK LINEAR MULTI-STEP METHODS*Chollom, J. P.; Ndam, J. N. & Kumleng, G. M.
Department of MathematicsUniversity of Jos, Nigeria *(Corresponding author) cholomj@unijos.edu.ng
ABSTRACTThe convergence, stability and order of Block linear Multistep methods have been determined in the past based on individual members of the block. In this paper, methods are proposed to examine the properties of the entire block. Some Block Linear Multistep methods have been considered, their convergence, stability and order have been determined using these approaches.
The search for higher order A-stable linear multi-step methods has been the interest of many numerical analyst and has been realized through either higher derivatives of the solution or by inserting additional off step points,supper future points and the likes.These methods are suitable for the solution of stiff differential equations which exhibit characteristics that place severe restriction on the choice of step size. It becomes necessary that only methods with large regions of absolute stability remain suitable for such equations. In this paper, high order block implicit multi-step methods of the hybrid form up to order twelve have been constructed using the multi-step collocation approach by inserting one or more off step points in the multi-step method. The accuracy and stability properties of the new methods are investigated and are shown to yield A-stable methods, a property desirable of methods suitable for the solution of stiff ODEs. The new High Order Block Implicit Multistep methods used as block integrators are tested on stiff differential systems and the results reveal that the new methods are efficient and compete favorably with the state of the art Matlab ode23 code.
A mathematical model for three-species interactions in a food chain, with the assumption that the interacting species are mobile, has been constructed using a combination of Holling's type III and the BD functional responses. Conditions for the onset of diffusive instability were determined. The results indicate the possibility of a stable coexistence of the three interacting species in form of stable oscillations under the reflecting boundary conditions. Habitat segregation also occurs under these conditions. However, under the absorbing boundary conditions, the species experience damped oscillations leading to their extinction. The effects of cross-diffusion of the intermediate and the toppredator were also examined.
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