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Extra resources for Computational Methods in Engineering Boundary Value Problems
28), gives 8(1/) and cj>(1/). In particular, we obtain 8(00) and cj>( (0). 3. Substituting 8( 00) and cj>( (0) into Eg. 30), the parameter s can be calculated. 4. The solution of N(1/) can be found by combining the solutions of the two initial value problems, according to Eg. 26). The concentration gradient on the surface of the probe, I, is given by 1= dN(O)/d1/ Using Eg. 31 ) Knowing s from step 3, I can be calculated. 3 shows I plotted as a function of (l + €)Nre/(l + fJ)R. This curve agrees with that given in .
5. 6. 7. , "The Numerical Treatment of Differential Equations," pp. 184-186, Springer-Verlag, New York, 1966. Lee, E. , Quasilinearization, nonlinear boundary value problems and optimization, Chern. Eng. , 21, 183-194 (1966). , and A. J. Sabadell, Electrostatic probe measurements in solid-propellant rocket exhausts, AIAA J. 8, 895-901 (1970). Moore, F. , "Theory of Laminar Flow," p. 127, Princeton Univ. Press, Princeton, New Jersey, 1964. , Sandwich beam analysis, J. Appl. Mech. 39, 773-778 (1972).
5) To convert the above boundary value problem to an initial value problem, the so-called adjoint system of equations to Eq. 1) will be introduced. The definition of the adjoint equations of Eq. 6) where the superscript T on A denotes the transposed matrix and the column matrix X is defined by We next differentiate the product XTy with respect to the independent variable, which gives ; (XTy) = XTy + XTy Substituting Y and XT from Eqs. 6) into Eq. 8) where the property that AIT = A is applied. The first and the third terms on the right-hand side of Eq.
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