Higher Derivative Block Method with Generalised Steplength for Solving First-Order Fuzzy Initial Value Problems
DOI:
https://doi.org/10.31436/iiumej.v24i1.2380Keywords:
Fuzzy Initial Value Problem, First-Order, Generalised Steplength, Block Method, Higher Derivative, Charging and Discharging of CapacitorAbstract
Block methods have been adopted in studies for solving first and higher order differential equations due to its impressive accuracy property. Taking a step further to improve this accuracy, researchers have considered the inclusion of higher-derivative terms in the block method, although this has been limited to the presence of one higher-derivative term in previous studies. Hence, this article aims at better accuracy by introducing two higher-derivative terms in the block method. In addition, this article presents a scheme with generalised step length such that there is flexibility on the choice of step length when developing the block method. The generalised step length scheme is adopted to develop a three-step block method for solving first-order fuzzy initial value problems. Its properties to ensure convergence and to show the region of absolute stability is investigated, and problems relating to charging and discharging of capacitor are considered. The absolute error shows the impressive accuracy of the three-step block method including obtaining the same values as the exact solution. Therefore, in addition to the new generalised algorithm presented in this article, a new three-step method for solving linear and nonlinear first order fuzzy initial value problems is presented.
ABSTRAK: Kaedah blok digunakan dalam banyak kajian untuk menyelesaikan persamaan pembezaan peringkat pertama dan peringkat tinggi kerana sifat ketepatannya yang baik. Bagi meningkatkan ketepatan ini, penyelidik telah mengambil kira dengan memasukkan terbitan peringkat tinggi dalam kaedah blok, walaupun ini terhad pada satu sebutan terbitan peringkat tinggi dalam kajian sebelum. Oleh itu, kajian ini bertujuan bagi mendapatkan ketepatan yang lebih baik dengan memperkenalkan dua sebutan terbitan peringkat tinggi dalam kaedah blok. Tambahan, kajian ini memperkenalkan skema dengan panjang-langkah kaki biasa supaya terdapat kebolehlenturan pada pilihan langkah semasa membangunkan kaedah blok. Skema ini diadaptasi bagi membangunkan kaedah blok tiga-langkah bagi menyelesai masalah nilai awal peringkat pertama secara rawak. Ciri-ciri terperinci dikaji bagi memastikan penumpuan lingkungan kestabilan mutlak, dan masalah berkaitan pengecasan dan nyahcas kapasitor juga turut diambil kira. Ralat mutlak menunjukkan ketepatan yang mengkagumkan pada kaedah blok tiga-langkah termasuk mendapatkan nilai yang sama seperti penyelesaian. Oleh itu, tambahan pada algoritma ini, kaedah tiga-langkah bagi menyelesaikan linear dan tidak linear pada masalah nilai awal peringat pertama secara rawak diperkenalkan.
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