mws_gen_ode_ppt_finitedifference(1)

# mws_gen_ode_ppt_finitedifference(1) - Finite Difference...

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1/10/2010 http://numericalmethods.eng.usf.edu 1 Finite Difference Method Major: All Engineering Majors Authors: Autar Kaw, Charlie Barker http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates

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Finite Difference Method http://numericalmethods.eng.usf.edu
http://numericalmethods.eng.usf.edu 3 Finite Difference Method An example of a boundary value ordinary differential equation is " 0030769 . 0 ) 8 ( , " 008731 . 0 ) 5 ( , 0 1 2 2 2 = = = + u u r u dr du r dr u d The derivatives in such ordinary differential equation are substituted by finite divided differences approximations, such as x y y dx dy i i + 1 ( ) 2 1 1 2 2 2 x y y y dx y d i i i + +

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http://numericalmethods.eng.usf.edu 4 Example Take the case of a pressure vessel that is being tested in the laboratory to check its ability to withstand pressure. For a thick pressure vessel of inner radius a and outer radius b, the differential equation for the radial displacement u of a point along the thickness is given by 0 1 2 2 2 = + r u dr du r dr u d The pressure vessel can be modeled as, ( ) 2 1 1 2 2 2 r u u u dr u d i i i + + r u u dr du i i + 1 Substituting these approximations gives you, ( ) 0 1 2 2 1 2 1 1 = + + + + i i i i i i i i r u r u u r r u u u ( ) ( ) ( ) 0 1 1 1 2 1 1 1 2 2 2 1 2 = + + + + i i i i i i u r u r r r r u r r r
http://numericalmethods.eng.usf.edu 5 Solution Step 1 At node " 5 , 0 0 = = = a r i " 0038731 . 0 0 = u Step 2 At node " 6 . 5 6 . 0 5 , 1 0 1 = + = + = = r r r i ( ) ( ) ( )( ) ( ) ( )( ) 0 6 . 0 6 . 5 1 6 . 0 1

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mws_gen_ode_ppt_finitedifference(1) - Finite Difference...

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