Heat conduction equation is approximated by the method:

Question

Heat conduction equation $u_{1} = u_{xx}$ is approximated by the method:

$u_{m}^{n+1}-u_{m}^{n-1} = frac{2k}{h^2}delta_{k}^{2} u_{m^.}^{n}$

Find the order of the method and investigate the stability of this method using Von Neumann method.

21 Mar 2023

Answer :

Word Count : 363

To find the order of the given method, we can use Taylor series expansions to write:

$u_{m}^{n+1} = u_{m}^{n-1} + 2k u_{xxm}^{n} + 2k^2/2 u_{xxxxm}^{n} + O(k^3)$

$u_{m}^{n-1} = u_{m}^{n+1} - 2k u_{xxm}^{n} + 2k^2/2 u_{xxxxm}^{n} + O(k^3)$

Substituting these into the given method, we get:

$u_{m}^{n+1} - u_{m}^{n-1} = 4k u_{xxm}^{n} + ______ ______ _______ ____ _______ __________ __________ ____ _________ _________. __________ _____ _______ _____ ___ _____. _________ ________ __________ _____ __________ _____ _____ ________ __________ _______ ________ ______. ____ _________ __________ ________ _________ ________ _________ ______ _______. ___ _______ _______ _________ ______. _____ ________ _____ __________ _______ ____ _____ ______ _________ _______ ____. ______ __________ _______ ____ ______ ___ _________ _________ ________ _____. _______ _______ _______ ___ _______ ________ ___ _________ ________. __________ ____ _______ __________ _______. _________ ____ __________ ____ _________ _________ ____ ______ _________ ________ _______. ________ _______ ____ ______ _______ ___ ________ _______ _________ ________ __________. ___ _______ ____ _______ ______ ________ ___ _________ ____ ______ __________ ______. __________ _____ ____ __________ ______. ________ _____ ____ _________ ____ _________. ___ _______ ________ ______ _______. __________ _____ _______ __________ _____ _______ _____ _____ _____ ________. _____ _________ ___ _________ __________ ______ _________. _________ _______ ______ _________ __________. _________ ___ ________ _______ _____ __________ _______. _____ __________ _____ ______ _________. _______ ______ _______ _____ ____ ____ ____. ____ _________ _____ ________ __________ ________ _______ ___ _________ ___. _______ ____ __________ ______ ______ ______ ___ ____ _______ ______. ___ ____ __________ ________ ____ ______ ________. ____ _________ ______ ___ ______ ___ ____. _________ ______ ______ ______ ________ __________ ______ ____ ____ _____ __________ ________. _____ _______ _______ __________ _________ _________ ________ _______ ______. _______ _________ _______ _______ ___ ____ ___ ___ ____ ____ _______ __________. _____ _______ ______ _____ __________. ________ _______ ___ _____ ______ ____ ____ ____ _________ _____. __________ ______ ________ ________ __________. ____ ________ _________ _______ ____ __________ ___ __________ __________ _______ _______ _________. _____ ____ __________ ___ _______. ___ ____ _______ _________ ________. _____ __________ _________ _________ ___ ___ ________ ____ __________ ____ _______. ___ ____ ____ ______ ____ ______ ______ _________. _________ _________ __________ ______ _______ _______ _________. <a class=$ Get Full Answer on WhatsApp

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