Question

this is numerical analysis. Please do a and b

4. Consider the ordinary differential equation 1'(x) = f(x, y(x)), y(ro) = Yo. (1) (a) Use numerical integration to derive the trapezoidal method for the above with uniform step size h. (You don't have to give the truncation error.) (b) Given below is a multistep method for solving (1) (with uniform step size h): bo +1 = 34 – 2n=1 + h (362. Yn) = f(n=1, 4n-1)) What is the truncation error for (2)? Is the method consistent? Is it stable? Is it convergent? If convergent, determine the order of convergence.

Answer 1

0 consider the following Initial Value Problem (IVP) y '(x) = f(x, y (x)), Y (Ho) - Yo To derive trapezoidal rule with uniform step-length h Let {20, 21, any be given data points such that h = Xiti-xi i-o.ton-1 be uniform step length consider, y '(x) = f(x, y(x)) On Integrating the differential equation from anto ant! y (n+1)- y (x n ) = f (x, y(x)) da to Let P(x) be an interpolating polynomial at xxn and x=ant. Then By Lagranges Interpolation formula P(x) = x-xnti get anti xn-Art fran, y(x)) tx-xn f(anti, y Cent)) anti-xn h h P(x) = anti-a f(xn, y (an)) + X-*f(anto, Ylanti) substituting this values in weget, y (anti)-Y(x) ba (f Can, 464mb) + f(anti, Ycun+ 1))) Let Ynt i = y(anti) and Yn = Y(un) y Conti) = Yati= Ynt th (f(xn, In)) + f(x1+1, Indr))) is required Trapezoidal rule. for given IVP 6 Given lineas multistep method is, Yn+1=3Yn - 2yn-1 the C f (an, Yn)-3f(xn-4, Yn 1)

& as= -1 and S=0 bo ½ b, - To find: 1 Truncation enor (1) Is it Stable 1 Is it consistent (W) Je it convergent And Order of convergence A Genecal lineas Multi-step method is given by a. Yn take Yn 1 t. 9 Inrichi ayn-e = h (befut the fore) ato so we take ak= 1 We know that A linear, multistep method is consistent if and only if (bg= At Ekat saj 9-0 Note that do toh+ 2-3 = 1 and bot bi = -1 = 2-3=-1 Given method is consistent. local truncation enor is 5 An ² you) where. Me (oxi-1, Hitl) to check this method is stable Consider one characteristic polynomial 3(27=22-322 862)= 2²-32 +2. which has roots) 2 = 1, 2 and a = 1 is the only Dot with modulus tie. 121=1 this system is Highly Stable (Corstrongly stable). method is convergent (as 2 = 1 is the only roots with 12 121-1 From (1) (a, a, 92) =(2, -3, 1) and (bo, b, bz) = (0, 2, 2)

To find order of the scheme Thus, do ao tata2 = 0 dy--bo+(9,-bi)+(29,-b2)=(-3+3 )+(2-4) de = 10,- be) +(203-253)=(+2+3)+244) di=9 + = 1 to Therefore the order of the scheme is 2