Convergence of the Variable Order and Variable Stepsize Direct Integration Methods for the Solution of the Higher Order Ordinary Differential Equations

The DI methods for directly solving a system ofa general higher order ODEs are discussed. The convergence of the constant stepsize and constant order formulation of the DI methods is proven first before the convergence for the variable order and stepsize case

Convergence Proof of Some Generalised Backward Differentiation (GBDF) Methods to Solve the General Second Order Ordinary Differential Equations (ODE).

Methods for the solution of the general second order ODE are discussed. Stiffness and convergence are also defined. Finally proof of convergence is given for certain cases of the GBDF methods for stiff pro blems

Organogenesis and Somatic Embryogenesis from Scalps of Musa Spp. Cv. Tanduk

This study was carried out to establish reliable and efficient regeneration protocols through organogenesis and somatic embryogenesis from scalps of Musa spp. cv. Tanduk. Scalps are highly proliferated meristems which are ideal materials for organogenesis and embryogenesis in Musa spp. Shoot-tips of Musa spp. Cv. Tanduk were cultred on a modified MS (Murashige and Skoog, 1962) medium containing various concentration of 0, 10, 50 and 100 pM BAP with 1.0 pM IAA added to each BAP concentration. The study showed a reasonably high percentage of scalp formation on MS medium supplemented with 100 pM of BAP and 1.0 pM IAA within 16 to 28 weeks after first culture. BAP at 50 pM produced poor quality scalps and 10 pM BAP resulted in higher proliferation of shoot buds. MS without BAP produced only a single elongated shoot from the cultured shoot tips. Subculture was performed on 4 weeks interval. Histological study was done following subculture of the proliferated shoot tip on 100 pM BAP treatment. Continuous changes were observed starting from a dome-shaped apical meristem covered by layers of leaf primordia to a zone of high division activity. This resulted in the proliferation of meristematic clumps aggregated in a cauliflower-like structure (scalps). The scalps appeared as undulating zone with h o w s separating the hdges of naked tiny meristems. Regeneration of shoots fiom scalps was tested on different concentrations of 1 .O, 2.5 and 5.0 pM BAP incorporated into MS basal medium. Treatment with 2.5 pM BAP produced an average of 40.63 shoots, the highest number of shoot-buds after 8 weeks of culture. The highest mean shoot-bud height attained after 8 weeks was 2.19 cm on 1.0 pM BAP. Shoots obtained from scalps were rooted on half strength and full strength MS media and on MS medium supplemented with 1.0, 5.0 and 10 pM IBA. Full-strength MS medium with 5.0 pM IBA produced the highest number of roots per explant (15.08). Meanwhile, the longest root length (1 1.07 cm) was attained in half strength MS medium. However, there was no significant difference root length observed between half-strength MS, full-strength MS and MS with 1.0 pM IBA. Plantlets produced fiom the scalps through organogenesis were acclimatized under 50% shade in four types of growing medium comprising of sand, peat, sand + top soil + goat dung (3:2: 1 vlv) and top soil + sand (1 : 1 v/v). After six weeks, the highest survivability (77.5%) was obtained in medium consisting of top soil + sand + goat dung. This was followed by 67.5 % survivability in both sand and peat media The lowest survivability (65%) was recorded fiom medium containing a mixture of top soil + sand. Plants acclimatised in medium containing top soil + sand + goat dung were observed to be morphologically the best with very green broad leaves and vigorous sterns. Multiplication of shoots from shoot-tip of banana cv. Tanduk was also investigated. The excised shoot-apices were cultured on MS basal medium supplemented with 0, 5, 10, 15 and 20 1M BAP in combination with 1.5 pM IAA in each. It was observed that BAP concentration over subculture cycles affected the mean number of shoots produced from the shoot tips and the mean shoot height (cm) attained. Shoot formation increased with increase in BAP concentrations and vice versa for shoot height. The highest mean number of shoot-buds (17.47) was produced on 20 pM BAP. The mean number of shoot-buds increased with subsequent subcultures at all levels of BAP tested, reaching the highest mean (27.53) at the 20& week of subculture. The highest mean height (2.43 cm) was obtained on MS medium supplemented with the lowest BAP concentration of 5 pM. Meanwhile, the mean shoot-bud height also increased over subculture cycles, reaching 3.14 cm at the fifth subculture (20 weeks). Two pathways of regeneration via somatic embryogenesis, starting fiom the scalps of Musa spp. cv. Tanduk were also investigated. Scalps were excised and cultured in liquid MS medium supplemented with 5.0 and 10 pM 2,4-D. Scalps placed on 10 pM 2,4-D died because of very high incidence of browning and necrosis. Meristematic globules were released fiom the scalps placed in liquid medium with 5.0 pM 2,4-D. The meristematic globules were either rounded, amoeboid or kidney shaped. Following transfer of the meristematic globules to fi-esh medium of the same composition for cell suspension establishment, yellowish embryogenic cells were formed at the periphery of the meristematic globules- The embryogenic cells were released as single cells into the medium which later gave rise to proembryo-like structures. However, these structures failed to regenerate into plantlets. The embryogenic response of scalps was also investigated on half strength MS semisolid medium containing 2.5, 5.0 and 7.5 pM 2,4-D with 1.0 pM zeatin. The study revealed that 5.0 pM 2,4-D with 1.0 pM zeatin produced the highest percentage of (48 %)embryogenic complexes. Histological sections of the embryogenic complexes induced on scalps of Musa spp. C.V. Tanduk revealed the presence of globular structures, pro-embryos and somatic embryos. Two regeneration media, Ganapathi et al. (1999) medium and CBte et al. (1996) medium were tested for plantlet regeneration fiom the embryogenic complexes. Both regeneration media tested did not exhibit high germination percentages which were 14- 17% on Cote et al. (1996) medium and 2-5% on Ganapathi et al. (1999) medium. Embryogenic cell suspensions were initiated from the scalps-derived embryogenic complex, in half strength MS medium supplemented with 5.0 pM 2,4-D and 1.0 pM zeatin. The study revealed that the embryogenic cell suspensions have high potential towards somatic embryogenesis. Homogeneous embryogenic lines were obtained and matured by plating onto filer paper discs wetted with semi-solid MS medium. The somatic embryos produced were placed on modified MS solid medium containing 0.5, 1.0, and 1.5 pM BAP for germination. The highest germination percentage of 52.60% was attained on medium with 0.5 pM BAP. However, there was no significant difference in the percentage of germination between the treatments. Histological observations on the different stages of somatic embryo development showed bipolar somatic embryos, matured somatic embryos with distinct shoots and root meristems, and two apical meristems originating from the same somatic embryo. Scalps were found to be appropriate in vitro materials for organogenesis and somatic embryogenesis. Somatic embryo regeneration through the establishment of embryogenic cell suspensions from the embryogenic complexes was the best for hture researches

Embedded singly diagonally implicit Runge-Kutta methods (4,5) in (5,6). For the integration of stiff systems of ODEs

In this paper, fourth order, five-stage embedded in fifth order six-stage Singly Diagonally Implicit Runge-Kutta (SDIRK) methods are derived, and their regions of stability are determined. Numerical results of these methods are presented and compared with the results obtained when the same systems are solved using Rabeh's third order three-stage embedded in fourth order four-stage SDIRK method. Also presented are the stability regions of the methods

Diagonally implicit hybrid method for solving special second order ordinary differential equations

This paper describes the derivation of a fifth-order diagonally implicit hybrid method. The method is zero dissipative and has phase-lag of order six. The method is compared with the existing hybrid method and the numerical comparisons carried out show that the new method improves the accuracy of the existing method for solving several special second order ordinary differential equations

Parallel block method for solving delay differential equations.

This paper considers the implementation of a two-point diagonally implicit block method for solving delay differential equations on parallel machines. The variable stepsize variable order block method produces two points simultaneously within a block. For greater advantage, formulae for the predictor-corrector scheme are represented in divided difference form. Performance of the parallelization is assessed in terms of speedup and efficiency

An application of Pontryagin's maximum principle in a linear quadratic differential game.

This paper deals with a class of two person zero-sum linear quadratic differential games, where the control functions for both players are subject to integral constraints. The duration of game is fixed. We obtain the necessary conditions of the Maximum Principle and also optimal control by using method of Pontryagin’s Maximum Principle. Finally, we discuss an example

An investigation of appropriateness relative to indigenous and modern housing in Libya

Space and the physical surroundings in which we live have an enormous bearing on our comfort and well being. Fortunate are those whose housing and urban environment are compatible with their behaviour and culture. However, relying on chance is insufficient when dealing with such an important aspect of life. The study aims to point the way to a long term solution to the Libyan housing crisis - a crisis not just of numbers, but of relevance of form.A detailed account of environmental, historical and socio- cultural influences on housing in Libya is given to set the scene for an examination of the present situation. Modern housing developments follow Western patterns, very different in construction and layout from indigenous forms. This background study gives a deeper insight to the social effects of such houses and pinpoints the ways in which they fail to meet the expectations and aspirations of their occupants.The focus is on comparing the space and form of new homes with the indigenous courtyard houses, whose development was arrested at the time of the Italian occupation. Because most of the courtyard housing areas are now in a run -down condition, modern and renovated forms presently used in other cultures, are described to show how they meet certain social and environmental needs also relevant to Libyan society.While progress should not be prevented, the direction and pace of change should be given thoughtful consideration. Libya's dramatic acquisition of wealth has caused her to lose the reins of development. Poverty and hardship of earlier years is not to be commended but it has been replaced by an alien infrastructure.Two clear points emerge. Indigenous courtyard housing provides a relevant, appropriate spatial structure but is sadly lacking in modern amenities. Modern western style housing is of insufficient quality to be durable, requires a high level of maintenance and management and fails to meet psychological and socio-cultural needs.Background notes and analysis of the case study are used to bring recommendations for the way ahead. As well as suggesting a structured network of research establishments to control standards and investigate needs, the recommendations include examples of houses which would provide a spatial structure capable of maintaining the culture and traditions of Libyan society

Implementation of four-point fully implicit block method for solving ordinary differential equations

This paper describes the development of a four-point fully implicit block method for solving first order ordinary differential equations (ODEs) using variable step size. This method will estimate the solutions of initial value problems (IVPs) at four points simultaneously. The method developed is suitable for the numerical integration of non-stiff and mildly stiff differential systems. The performances of the four-point block method are compared in terms of maximum error, total number of steps and execution times to the non-block method 1PVSO in [Z. Omar, Developing parallel block methods for solving higher order ODEs directly, Ph.D. Thesis, University Putra Malaysia, Malaysia, 1999]