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Additional resources for The Method of Weighted Residuals and Variational Principles: With Application in Fluid Mechanics, Heat and Mass Transfer
Warren, W. E. (1964). Bending of Rhombic Plates, AIAA J . 2, 166-168. Wasserman, M. , and Slattery, J. C. (1969). Creeping Flow past a Fluid Globule When a Trace of Surfactant Is Present, AZChE J. 15, 533-547. Chapter 3 Eigencalue ancl Initial-Vulue Problems in Heat ancl Mass Transfer One of the two most prolific areas of MWR application is transient heat transfer problems (the other prolific area being boundary layer flows). A great many of these applications are for one- or two-term approximations.
7) The weighted residual becomes jow,R(x, 0,) 1 dx = 0, k = 1 , 2 , . , N . 8) We next apply several criteria for comparison. For the first approximation 8, = x + A , ( x ~- x), 8,’ = 1 + A , ( ~ x- I), 8,” = 2 4 . 9) Apply the collocation method using the collocation point x = 4 because it is the midpoint of the interval. Another point could be chosen, but low-order approximations are likely to be better if the collocation points are distributed somewhat evenly throughout the region. 10) which determines A , .
Thus the first few functions are 1, x, y , x2 - y 2 , x3 - 3xy2, y 3 - 3yx2. Shih (1970) used the boundary collocation method to solve for the temperature distribution in a square column surrounding a heating cylinder. Details of a computer code useful in these problems is in Davis (1962). Many solutions in two-dimensional heat transfer problems can be deduced from solutions for the velocity in ducts [Eq. 11 or from solutions to the torsion problem [Eq. 19) with A = 21. A number of these solutions, as well as other references to boundary collocation in the field of elasticity (where it is called point matching) are contained in Sattinger and Conway (1965) and Leissa and Neidenfuhr (1966) and the references cited by them.