## Advanced Topics in Applied Mathematics: For Engineering and by Sudhakar Nair

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. complicated themes in utilized arithmetic covers 4 crucial utilized arithmetic issues: Green's services, essential equations, Fourier transforms, and Laplace transforms. additionally integrated is an invaluable dialogue of themes equivalent to the Wiener-Hopf approach, Finite Hilbert transforms, Cagniard-De Hoop process, and the right kind orthogonal decomposition. This publication displays Sudhakar Nair's lengthy lecture room adventure and contains various examples of differential and imperative equations from engineering and physics to demonstrate the answer tactics. The textual content comprises workout units on the finish of every bankruptcy and a options guide, that is to be had for teachers.

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**Example text**

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. For these integrals to exist, certain conditions on the decay of f at inﬁnity are required. Of course, in bounded domains, g∞ does not satisfy the boundary conditions, and we have to resort to other methods.

14 Find the Green’s function for x2 u − xu + u = f (x), u(0) = 0, u(1) = 0. 15 Using the self-adjoint form of the differential equation x2 u + 3xu − 3u = f (x), u(0) = 0, u(1) = 0, ﬁnd the Green’s function and obtain an explicit solution when f (x) = x. 16 Solve the equation x2 u + 3xu = x2 , u(1) = 1, u(2) = 2, u(0) = 0, u (1) = 0, using the Green’s function. 17 For the problem Lu = u + u = 0, obtain the adjoint system. Solve the eigenvalue problems, Lu = λu, L∗ v = λv, and show that their eigenfunctions are bi-orthogonal.

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.