3. Develop frst order approximation Foruyy i jusing the pointsuij,u+1,u+2nonuniFormlyspaced.Leth1=yj+1−yjandh2=yj−yj−1,thenuyy i j=Au+Bu+1+Cu−1withA, B, Cto be determined. Now take Taylor series expansionsu+1=u+h1uyij+h212uyy i j+h316uyyy i j+···u−1=u−h2u+h222uyy i j−h326uyyy i j+SoAu++1+−1=(A+B+C)u+(Bh1−Ch2)uBh212+Ch222)uyy i jBh316−Ch326)uyyy i j±···Compare coeﬃcients withuyy i jto getA+B+C=01−2Bh212+Ch222=1This system oF 3 equations can be solved ForA, B, Cto get From the secondB=Ch2h1Plugging in the third and solve For
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This note was uploaded on 12/22/2011 for the course MAP 2302 taught by Professor Bell,d during the Fall '08 term at UNF.