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

https://faculty.kfupm.edu.sa/math/fzaman/ijhmt.pdf

Initial inverse problem in heat equation with Bessel operator

We investigate the inverse problem in the heat equation involving the recovery of the initial temperature front measurements of the final temperature. This problem is extremely ill-posed and it is believed that only information in the first few modes can be recovered by classical methods. We will consider this problem with a regularizing parameter which approximates and regularizes the heat conduction model

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Investigation of the Initial Inverse Problem in the Heat Equation

The classical inverse problem of recovering the initial temperature distribution from the final temperature distribution is extremely ill-posed. We propose a class of numerical schemes based on positivity-preserving Pade approximations to solve initial inverse problems in the heat equation. We also utilize a partial fraction decomposition technique to solve the problem more efficiently when higher order Pade approximations are used. We apply the proposed numerical schemes on the parabolic heat equation. Our aim is to model the problem as a direct problem and use our numerical schemes to recover the initial profile in a stable and efficient way.

Solution of the Initial Inverse Problems in the Heat Equation Using the Finite Difference Method with Positivity-Preserving Padé Schemes

A new approach for generating approximate analytic solutions of transient nonlinear heat conduction problems is presented. It is based on an effective combination of Lie symmetry method, homotopy perturbation method, finite element method, and simulation based error reduction techniques. Implementation of the proposed approach is demonstrated by applying it to determine approximate analytic solutions of real life problems consisting of transient nonlinear heat conduction in semi-infinite bars made of stainless steel AISI 304 and mild steel. The results from the approximate analytical solutions and the numerical solution are compared indicating good agreement.

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Approximate Analytic Solutions of Transient Nonlinear Heat Conduction with Temperature-Dependent Thermal Diffusivity

Lie symmetry method is applied to analyze a nonlinear elastic wave equation for longitudinal deformations with third-order anharmonic corrections to the elastic energy. Symmetry algebra is found and reductions to second-order ordinary differential equations (ODEs) are obtained through invariance under different symmetries. The reduced ODEs are further analyzed to obtain several exact solutions in an explicit form. It was observed in the literature that anharmonic corrections generally lead to solutions with time-dependent singularities in finite times. Along with solutions with time-dependent singularities, we also obtain solutions which do not exhibit time-dependent singularities.

Khalid Masood

Papers

Initial inverse problem in heat equation with Bessel operator

Investigation of the Initial Inverse Problem in the Heat Equation

Solution of the Initial Inverse Problems in the Heat Equation Using the Finite Difference Method with Positivity-Preserving Padé Schemes

Approximate Analytic Solutions of Transient Nonlinear Heat Conduction with Temperature-Dependent Thermal Diffusivity

Symmetry solutions of a nonlinear elastic wave equation with third-order anharmonic corrections