Differential equations play a prominent role in engineering and research fields in modeling engineering structures, describing important phenomena, and simulating mathematical behavior of engineering dynamical systems. Because of the increasing complexity of modern engineering systems, computationally efficient methods are demanded for solving these differential equations. In order to meet this challenge, this thesis presents two efficient algorithms for solving two types of differential equations: a one-dimensional heat equation with variable properties, and a one-dimensional parabolic equation, both of which are very popular and important in current engineering systems. In this study, the two equations were successfully solved using He’s variational iteration technique, and efficient algorithms have been developed. Detailed procedures for developing these algorithms are presented.
At first, a unique algorithm for solving the one-dimensional heat equations was developed by using the iteration variational approach. The accuracy of this algorithm was found by comparing the obtained solutions with the exact ones.
And similarly, using variational iteration approach, another efficient algorithm for solving the one-dimensional parabolic equation was developed. Three illustrative numerical problems were solved and the obtained results were compared with those yielded from the Adomian decomposition method (ADM) to verify the efficiency and accuracy of the developed algorithm.
With the encouraging results obtained from this study, it is expected that, in the future, developed algorithms can be extended to solve other differential equation systems, thus achieving a broader applicability in engineering and other research fields.
|Commitee:||Lee, Jim, Simon, William E.|
|School:||University of Louisiana at Lafayette|
|School Location:||United States -- Louisiana|
|Source:||MAI 52/06M(E), Masters Abstracts International|
|Subjects:||Applied Mathematics, Mechanical engineering|
|Keywords:||Closed form solutions, Differential equation, He's variational iteration method, Heat equation, Numerical algorithm, One-dimensional parabolic equation|
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