16/06/2015 MNT OREN, EE Numerical Methods for Differential Equations Department of Biomedical Engineering Department of Materials Science & Nanotechnology Engineering TOBB University of Economics and Technology Ankara - TURKEY Dr. Ersin Emre Oren MNT 521 Computational Methods in Bio- & Nano-technology

16/06/2015 MNT OREN, EE Runge-Kutta Methot: k 1 k 1 =  x. f(x i, y i ) k 2 +k 1 k 2 =  x f(x i +  x /2, y i +k 1 /2) y(x) =? P1P1P1P1 +k 1 y i +k 1 /2 P2P2P2P2 +k 2 y i +k 2 /2 P3P3P3P3 k 3 +k 2 k 3 =  x f(x i +  x /2, y i +k 2 /2) +k 3 y i +k 3 P4P4P4P4 k 4 +k 3 k 4 =  x f(x i +  x, y i +k 3 ) y i+1 Numerical Solution of Differential Equations

16/06/2015 MNT OREN, EE Finite Difference Methods: First Derivative yy y i+1 Forward difference Backward difference y i-1 yy Centered difference Numerical Solution of Differential Equations

16/06/2015 MNT OREN, EE Finite Difference Methods: Second Derivative yy y i+1 Forward difference Backward difference y i-1 yy xx Second derivative: Finite Difference