Download Advanced Topics in Applied Mathematics - For Engineering and by Sudhakar Nair PDF

By Sudhakar Nair

This booklet is perfect for engineering, actual technology, and utilized arithmetic scholars and pros who are looking to increase their mathematical wisdom. complex themes in utilized arithmetic covers 4 crucial utilized arithmetic issues: Green's services, critical equations, Fourier transforms, and Laplace transforms. additionally integrated is an invaluable dialogue of issues equivalent to the Wiener-Hopf strategy, Finite Hilbert transforms, Cagniard-De Hoop approach, and the correct orthogonal decomposition. This e-book displays Sudhakar Nair's lengthy school room event and contains quite a few examples of differential and vital equations from engineering and physics to demonstrate the answer tactics. The textual content contains workout units on the finish of every bankruptcy and a recommendations handbook, that's to be had for teachers.

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1 Example: Steady-State Heat Conduction in a Plate Consider an infinite plate under steady-state temperature distribution with a heat source distribution, q(x, y). 176) where k is the conductivity. Using the two-dimensional Green’s function, the solutions is written as T(x, y) = − 1 4πk ∞ ∞ −∞ −∞ q(ξ , η) log[(x − ξ )2 + (y − η)2 ] dξ dη. 177) Usually, the source is limited to a finite area, and the limits of the above integral will have finite values. If the heat source has a circular boundary, polar coordinates may be more convenient.

8. Two-dimensional domain. With p = 1 and q = 0, the Sturm-Liouville equation becomes the Poisson equation ∇2u = f . 172) We could apply the above integration using the Gauss theorem for the two-dimensional (2D) Sturm-Liouville equation (see Fig. 8). This results in dg 1 dg . 174) with the exact Green’s function for the infinite domain, g∞ = 1 log r, 2π r = {(x − ξ )2 + (y − η)2 }1/2 . 175) Now we have exact Green’s functions for the Laplace operators in 2D and 3D infinite spaces. To obtain the solution u in terms of g∞ , we need to compute the integrals of f multiplied by g over the whole space.

Assume λn and µn are the sequences of eigenvalues associated with these eigenfunctions. That is Lun = λn un , L∗ vn = µn vn . 140) We assume that each of the sequence of eigenvalues are distinct and the eigenfunctions are complete. Then all of the v’s cannot be orthogonal to a given un . Let us denote by vn one of these v’s that is not orthogonal to un . 141) which shows µn = λn . The two operators, L and L∗ , have the same eigenvalues, Lun = λn un , L∗ vm = λm vm . 142) Forming inner products of the first equation with vm and the second with un , we get vm , Lun − un , L∗ vm = (λn − λm ) vm , un = 0.

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