Advanced Mathematical Methods for Scientists and Engineers

The goal of this final chapter is to show how the boundary value problems of mathematical physics can be solved by the methods of the preceding chapters. This will be done by solving a variety of specific problems that illustrate the principal types of problems that were formulated in Chapter 7. Additional applications are developed in the Exercises. The primary solution method is Fourier’s method of separation of variables and the associated Sturm-Liouville theory of Chapter 8.

Boundary Value Problems of Mathematical Physics

We study the spatiotemporal sampling of a diffusion field generated by K point sources, aiming to fully reconstruct the unknown initial field distribution from the sample measurements. The sampling operator in our problem can be described by a matrix derived from the diffusion model. We analyze the important properties of the sampling matrices, leading to precise bounds on the spatial and temporal sampling densities under which perfect field reconstruction is feasible. Moreover, our analysis indicates that it is possible to compensate linearly for insufficient spatial sampling densities by oversampling in time. Numerical simulations on initial field reconstruction under different spatiotemporal sampling densities confirm our theoretical results.

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Sampling and reconstructing diffusion fields with localized sources

A comprehensive survey of the literature in the area of numerical heat transfer (NHT) published between 2000 and 2009 has been conducted Due to the immenseness of the literature volume, the survey

Literature Survey of Numerical Heat Transfer (2000–2009): Part II

Heat transfer—A review of 2004 literature

Inverse scattering in multilayer inverse problem in the presence of damping

Investigation of the direct and inverse solutions for torsional waves propagating in a cylinder

A one-dimensional inverse problem in a medium with constant-background properties is studied in the presence of damping. The inversion formula is obtained using Born's approximation and high frequency assumption. This formula is tested for a medium which contains a layer giving rise to a single step-like change in the wavespeed and damping parameters. The effects of background damping and of bandlimited data are also discussed by taking different values of background damping and bandlimiting.

Testing of the inverse problem with constant-background in the presence of damping in a single layer

We consider a problem of identification of physical properties of the Earth using the damped wave equation, based on the linearized inversion associated with Born's inversion theory. We assume that damping and sound speed are well approximated by the background plus the perturbation. Application of the method leads to a linear integral equation involving variations in sound speed and damping. Our aim is to recover these variations in velocity and damping, what in turn yields a map of the interfaces in the interior of the Earth. We consider the three-dimensional inverse problem of determining three-dimensional variations in the propagation speed and damping by considering the damped wave equation. We exploit the high-frequency character of seismic data to simplify the problem.

Khalid Masood

Papers

Inverse scattering in multilayer inverse problem in the presence of damping

Investigation of the direct and inverse solutions for torsional waves propagating in a cylinder

Testing of the inverse problem with constant-background in the presence of damping in a single layer

Recovery of medium properties in three dimensions with three-dimensional parameters variability