id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
american_scientific_journal-1248	Hassan, Eltigani Ismail	Solution of Laplace`s Equation Using Numerical Methods	2016	15	.pdf	application/pdf	3145	130	65	Volume 16, No 1, pp 178-192 for i = 2:n for j = 2:m TN(i,j)=(T(i-1,j)+T(i+1,j)+beta^2*(T(i,j-1)+T(i,j+1)))/denom; err(i,j) = abs(TN(i,j)-T(i,j)); end; end; T = TN; k = k + 1; errmax = max(max(err)); if errmax< epsilon [X,Y] = meshgrid(x,y); figure(2);contour(X,Y,T',20);xlabel('x');ylabel('y'); title('Laplace equation solution - Dirichlet boundary conditions - Explicit'); figure(3);surfc(X,Y,T');xlabel('x');ylabel('y');zlabel('T(x,y)'); title('Laplace equation solution - Dirichlet boundary conditions - Explicit'); fprintf('Convergence achieved after %i iterations.\n',k); Hence the total gain of heat by rectangular element per second 179 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 16, No 1, pp 178-192 = −𝑘𝛼𝛼𝜕𝑥 �𝜕𝑢 𝜕𝑦 � 𝑦 − 𝑘𝛼𝛼𝜕𝑦 �𝜕𝑢 𝜕𝑥 � 𝑥 + 𝑘𝛼𝛼𝜕𝑥 �𝜕𝑢 𝜕𝑦 � 𝑦+𝜕𝑦 + 𝑘𝛼𝛼𝜕𝑦 �𝜕𝑢 𝜕𝑥 � 𝑥+𝜕𝑥 = 𝑘𝛼𝛼𝜕𝑥 � �𝜕𝑢𝜕𝑥�𝑥+𝜕𝑥 −�𝜕𝑢𝜕𝑥�𝑥 𝜕𝑥 + �𝜕𝑢𝜕𝑦�𝑦+𝜕𝑦 −�𝜕𝑢𝜕𝑦�𝑦 𝜕𝑦 � Also the rate of gain of heat by the element = 𝜌𝜌𝜕𝑥𝜕𝑦𝛼𝛼𝑠 𝜕𝑢 𝜕𝑡 Thus equating (1) and (2) = 𝑘𝛼𝛼𝜕𝑥𝜕𝑦 � �𝜕𝑢𝜕𝑥�𝑥+𝜕𝑥 −�𝜕𝑢𝜕𝑦�𝑥 𝜕𝑥 + �𝜕𝑢𝜕𝑥�𝑦+𝜕𝑦 −�𝜕𝑢𝜕𝑦�𝑦 𝜕𝑦 � = 𝜌𝜌𝜕𝑥𝜕𝑦𝛼𝛼𝑠 𝜕𝑢 𝜕𝑡 (3)	cache/american_scientific_journal-1248.pdf	txt/american_scientific_journal-1248.txt