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1. To obtain a solution through superposition of solutions, describe the number of non-homogeneous conditions per sub-problem. 2.    Discuss...

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Mechanical Engineering

1. To obtain a solutionthrough superposition of solutions, describe the number ofnon-homogeneous conditions per sub-problem.

2.   Discuss the relevance of shape factors.

3.   Discuss the requirements for using the superposition method ofsolution.

4.   Describe what is meant by eigen-problem solution in heattransfer.

5.   Discuss what values a separation constant can attain during theseparation of variables method of solution.

Answer & Explanation Solved by verified expert
4.3 Ratings (551 Votes)
1 Assume we have two different solutions xttxtt and xttxtt to an ODE of the form dxdtatxt0dxdtatxt0 For the two curves to cross each other we have to have t0t0t0t0 and ddtttt0ddtttt0ddtttt0ddtttt0 for some t0t0 With ddtatt0ddtatt0 and ddtatt0ddtatt0 we see that this is impossible Because the curves cannot cross they must be uniquely determined by their value for some tt0tt0 Now in a second order ODE like d2xdt2atdxdtbtx0d2xdt2atdxdtbtx0 two solutions can cross each other But when we assume two solutions with t0t0t0t0 and ddtttt0ddtttt0ddtttt0ddtttt0 and d2dt2ttt0d2dt2ttt0d2dt2ttt0d2dt2ttt0 for some t0t0 we again see that this cannot be done So for two solutions tt and tt with t0t0t0t0 we see that the curves of ddtddt and ddtddt can never cross each other because when ddtddt and ddtddt are equal in t0t0 then their derivatives ddtddtddtddt and ddtddtddtddt must also be equal Thus curves of dxdtdxdt are uniquely    See Answer
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