By C.E. Weatherburn

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**Example text**

Sin X=Sin X COSt 27 THE DECOMPOSrriON MEFHODINSEVERALDIMENS/ONS We are dealing with a methodology for solution of physical systems which have a solution, and we seek to find this solution without changing the problem to make it tractable. The conditions must be known; otherwise, the model is not complete. If the solution is known, but the initial conditions are not, they can be found by mathematical exploration and consequent verification. :o An where the An are generated for the specific f(u).

Thus we will drop terms involving and x3 and higher terms. Then e -t e e =1-t+2 sinx= x x2 cosx=l-2 so that g==2(1-t+ ~}x)-2(1-2t+2e)x(l- ~) Then L-1 g == 0 to the assumed approximation. Hence u0 = x-tx U1 = L~'(ajat)A 0 + L~'(a 2 /ax 2 )u0 = xe/2 Thus the two-term approximation is C/J2 = u0 +u 1 = x- tx +xe/2 = (1- e + e/2 )x == e -t sin X Although we can calculate more terms using Um+t for m ~ 0, substitution verifies that u = e-1sin x is already the correct solution. If we need to recognize the exact solution, we can carry the series for g to a higher approximation to see the clear convergence to e- 1 sinx.

Sin X=Sin X COSt 27 THE DECOMPOSrriON MEFHODINSEVERALDIMENS/ONS We are dealing with a methodology for solution of physical systems which have a solution, and we seek to find this solution without changing the problem to make it tractable. The conditions must be known; otherwise, the model is not complete. If the solution is known, but the initial conditions are not, they can be found by mathematical exploration and consequent verification. :o An where the An are generated for the specific f(u).