The diagonally implicit 2-point block backward differentiation formulas (DI2BBDF) of order two, order three, and order four are derived for solving stiff initial value problems (IVPs). The stability properties of the derived methods are investigated. The implementation of the method using Newton iteration is also discussed. The performance of the proposed methods in terms of maximum error and computational time is compared with the fully implicit block backward differentiation formulas (FIBBDF) and fully implicit block extended backward differentiation formulas (FIBEBDF). The numerical results show that the proposed method outperformed both existing methods.
This paper focuses on the derivation of diagonally implicit two-point block backward differentiation formulas (DI2BBDF) for solving first-order initial value problem (IVP) with two fixed points. The method approximates the solution at two points simultaneously. The implementation and the stability of the proposed method are also discussed. A performance of the DI2BBDF is compared with the existing methods.
Malaysia is in the process of liberalising its electricity supply industry (ESI) further, with the second reform series announced in September 2018. If everything goes as planned, Malaysia would be the third country in the Association of Southeast Asia Nations (ASEAN) to have a fully liberalised ESI after the Philippines and Singapore. A number of initiatives have been in the pipeline to be executed and a lot more will be planned. At this juncture, it is important for Malaysia to look for the best practices and lessons that can be learnt from the experience of other countries that have successfully liberalised their ESIs. Being in the same region, it is believed that there is a lot that Malaysia can learn from the Philippines and Singapore. This paper therefore presents and deliberates on the chronological development of the countries’ progressive journeys in liberalising their ESIs. The aim is to discern the good practices, the challenges as well as the lessons learnt from these transformations. Analysis is being made and discussed from the following four perspectives; legislative framework, implementation phases, market components and impact on renewable energy penetration. Findings from this study would provide useful insight for Malaysia in determining the course of actions to be taken to reform its ESI. Beyond Malaysia, the findings can also serve as the reference for the other ASEAN countries in moving towards liberalising their ESIs.
In this research, a singly diagonally implicit block backward differentiation formulas (SDIBBDF) for solving stiff ordinary differential equations (ODEs) is proposed. The formula reduced a fully implicit method to lower triangular matrix with equal diagonal elements which will results in only one evaluation of the Jacobian and one LU decomposition for each time step. For the SDIBBDF method to have practical significance in solving stiff problems, its stability region must at least cover almost the whole of the negative half plane. Step size restriction of the proposed method have to be considered in order to ensure stability of the method in computing numerical results. Efficiency of the SDIBBDF method in solving stiff ODEs is justified when it managed to outperform the existing methods for both accuracy and computational time.
This paper proposed 2-point block backward differentiation formulas (BBDF) of order 3, 4, and 5 for direct solution of second order ordinary differential equations. These methods were derived via backward difference interpolation polynomial with two solutions are produced simultaneously at each step. All the three different orders of 2-point BBDF is implemented in variable order scheme. The scheme utilizes the local truncation error, which is generated by the single order of 2-point BBDF method. Numerical results are presented to illustrate the validity of the proposed scheme.
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