id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
cuesj-205	Nadir, Narmeen N.	Series Solution for Single and System of Non-linear Volterra Integral Equations	2020	3	.pdf	application/pdf	2384	111	67	Again, to find u” (a), differentiate equation (3) with respect to x and get �� � �� � �� � ��u x v x k x t u t dt K t u t u t dt t a t x t a t x ( ) ( , ( ), ( )) (9) use principle of (7) for both side of (9), we obtain � � � � � � �u x u K t u t u t dt t a t x ( ) ( , ( ), ( ))1 (10) Take integral again for both side of (10), we can rewrite (10) as, � �� � � �� ��u t u dt K z u z u z dzdx a x a z a x ( ) ( , ( ), ( ))1 (11) u x u a u x a x a K z u z u z dz a x ( ) ( ) ( ) ( ) ( , ( ), ( ))� � � � � ��1 Nadir:	cache/cuesj-205.pdf	txt/cuesj-205.txt