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3) a) Use the Taylor theorem for function U(x, t) with the step size(-Ax) (t, is held cons. b) Consider the second-order truncation error (0 (Ax)2) with the step size (-Ax) and then, obtain a finite difference approximation for the first-order derivative of U(x, t) respect x (Ux(x, t)).

3) a) Use the Taylor theorem for function U(x, t) with the step size(-Ax) (t, is held cons. b) Consider the second-order truncation error (0 (Ax)2) with the step size (-Ax) and then, obtain a finite difference approximation for the first-order derivative of U(x, t) respect x (Ux(x, t)).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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3) a) Use the Taylor theorem for function U(x, t) with the step size(-Ax ) (t, is held cons. b) Consider the second-order truncation error (0(Ax )²) with the step size (-Ax ) and then, obtain a finite difference approximation for the first-order derivative of U(x, t) respect x (Ux(x, t)). c) Consider the third-order truncation error (0 (Ax )³) with the step size Ax and also (-Ax) then, subtract them and finally obtain a finite difference approximation for the first-order derivative of U respect x (Ux(x, t)). d) Consider the fourth-order truncation error (O(Ax )4) with the step size Ax and also (-Ax ) then, add them and finally obtain a finite difference approximation for the second-order derivative of U respect x (Uxx(x, t)).
Transcribed Image Text:3) a) Use the Taylor theorem for function U(x, t) with the step size(-Ax ) (t, is held cons. b) Consider the second-order truncation error (0(Ax )²) with the step size (-Ax ) and then, obtain a finite difference approximation for the first-order derivative of U(x, t) respect x (Ux(x, t)). c) Consider the third-order truncation error (0 (Ax )³) with the step size Ax and also (-Ax) then, subtract them and finally obtain a finite difference approximation for the first-order derivative of U respect x (Ux(x, t)). d) Consider the fourth-order truncation error (O(Ax )4) with the step size Ax and also (-Ax ) then, add them and finally obtain a finite difference approximation for the second-order derivative of U respect x (Uxx(x, t)).
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