id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
iajs-2860	Ibrahim, Zahraa A. ; Hasan, Nabaa N. 	Approximation Solution of Fuzzy Singular Volterra Integral Equation by Non-Polynomial Spline	2023	8	.pdf	application/pdf	3320	167	75	, �̃�𝑟 be r-fuzzy subsets of 𝑋1, 𝑋2, … , 𝑋𝑟 , respectively, 𝑓 = X → 𝑌, 𝑦 = 𝑓(𝑥1, 𝑥2, … , 𝑥𝑟), then the extensions principles allow us to define a fuzzy set �̃� in Y by: �̃� = {(𝑦, 𝜇�̃�(𝑦))| 𝑦 = 𝑓(𝑥1, 𝑥2, … , 𝑥𝑟), 𝑥1, 𝑥2, … , 𝑥𝑟 ∈ X} where 𝜇�̃�(𝑦) = { sup 𝑀𝑖𝑛{𝜇𝐴1̃ (𝑥1), … , 𝜇𝐴�̃� (𝑥𝑟)}, 𝑓−1(𝑦) ≠ ∅ (𝑥1, 𝑥2, … , 𝑥𝑟) ∈ 𝑓−1(𝑦) 0 , 𝑜𝑡ℎ𝑒𝑟 𝑤𝑖𝑠𝑒 and 𝑓−1 is the inverse image of 𝑓 for 𝑟 = 1, the fuzzy extension principles, of course reduces to �̃� = 𝑓(�̃�) = 𝑓({(𝑦, 𝜇�̃�(𝑦))| 𝑦 = 𝑓(𝑥) , 𝑥 ∈ X} where 𝜇�̃�(𝑦) 5. NPS of FVIE with Abel's type kernel: Singular FVIE with Abel's type kernel [3] can be written in a general form as �̃�(𝑥, 𝑟) = 𝑓(𝑥, 𝑟) + ∫ 𝑢(𝑡,𝑟) √𝑥−𝑡 𝑥 0 𝑑𝑡. (26) Where 𝑟, 𝑥 ∈ [0,1] To solve equation (26), applying Laplace transformation to both sides, we have ℒ�̃�(𝑥, 𝑟) = ℒ𝑓(𝑥, 𝑟) +	cache/iajs-2860.pdf	txt/iajs-2860.txt