Ahmad Y. Al-Dweik

Wave equation on spherically symmetric Lorentzian metrics

Wave equation on a general spherically symmetric spacetime metric is constructed. Noether symmetries of the equation in terms of explicit functions of θ and ϕ are derived subject to certain differential constraints. By restricting the metric to flat Friedman case the Noether symmetries of the wave equation are presented. Invertible transformations are constructed from a specific subalgebra of these Noether symmetries to convert the wave equation with variable coefficients to the one with constant coefficients.

Symmetry analysis of wave equation on static spherically symmetric spacetimes with higher symmetries

Symmetry analysis of wave equation on all static spherically symmetric spacetimes admitting maximal isometry groups G10 or G7 or G6 is carried out. Symmetry algebras of the wave equation are found and their structural information-in the sense of Iwasawa decomposition-is obtained. Joint invariants of appropriate subalgebras are utilized to obtain many exact solutions of the wave equation on static spherically symmetric spacetimes.

A point symmetry based method for transforming ODEs with three-dimensional symmetry algebras to their canonical forms

We provide an algorithmic approach to the construction of point transformations for scalar ordinary differential equations that admit three-dimensional symmetry algebras which lead to their respective canonical forms.

Laser Pulse Heating of Surfaces and Thermal Stress Analysis

PREFACE.- ACKNOWLEDGEMENTS.- INTRODUCTION.- EQUILIBRIUM LASER PULSE HEATING AND THERMAL STRESS ANALYSIS.- INTRODUCTION.- STEP INPUT LASER PULSE HEATING.- STRESS FREE BOUNDARY AT THE SURFACE.- STRESS CONTINUITY BOUNDARY AT THE SURFACE.- TIME EXPONENTIALLY VARYING LASER PULSE HEATING.- STRESS FREE BOUNDARY AT THE SURFACE.- STRESS FREE BOUNDARY AND CONVECTION AT THE SURFACE.- STRESS BOUNDARY AT THE SURFACE.- ENTROPY ANALYSIS DUE TO THERMAL STRESS FIELD.- FINDINGS AND DISCUSSIONS.- STEP INPUT PULSE HEATING.- TIME EXPONENTIALLY VARYING LASER PULSE HEATING.- ENTROPY ANALYSIS DUE TO THERMAL STRESS FIELD.- ANALYTICAL SOLUTION OF CATTANEO AND THERMAL STRESS EQUATIONS.- INTRODUCTION.- SURFACE HEAT SOURCE CONSIDERATION.- STEP INPUT PULSE HEATING.- EXPONENTIAL PULSE HEATING.- VOLUMETRIC SOURCE CONSIDERATION.- STEP INPUT PULSE HEATING.- EXPONENTIAL PULSE HEATING.- ENTROPY ANALYSIS.- FINDINGS AND DISCUSSION.- SURFACE HEAT SOURCE CONSIDERATION.- VOLUMETRIC HEAT SOURCE CONSIDERATION.- ENTROPY GENERATION RATE.- ANALYTICAL TREATMENT OF HYPERBOLIC EQUATIONS FOR STRESS ANALYSIS.- INTRODUCTION.- FORMULATION OF ENERGY TRANSPORT IN METALLIC SUBSTRATES AT MICROSCOPIC LEVEL.- THERMAL STRESS FIELD: CONSIDERATION OF SURFACE AND VOLUMETRIC SOURCES.- SURFACE HEAT SOURCE CONSIDERATION.- VOLUMETRIC HEAT SOURCE CONSIDERATION.- THERMAL STRESS FIELD: TWO-DIMENSIONAL CONSIDERATION.- FINDINGS AND DISCUSSIONS.- SURFACE HEAT SOURCE CONSIDERATION.- VOLUMETRIC HEAT SOURCE CONSIDERATION.- TWO-DIMENSIONAL ANALYSIS.- CONCLUDING REMARKS.- EQUILIBRIUM HEATING.- CATTANEO HEATING MODEL AND THERMAL STRESSES.- NON-EQUILIBRIUM HEATING.

On the Linearization of Second-Order Ordinary Differential Equations to the Laguerre Form via Generalized Sundman Transformations

The linearization problem for nonlinear second-order ODEs to the Laguerre form by means of generalized Sundman transformations (S-transformations) is considered, which has been investigated by Duarte et al. earlier. A characterization of these S-linearizable equa- tions in terms of first integral and procedure for construction of linearizing S-transformations has been given recently by Muriel and Romero. Here we give a new characterization of S- linearizable equations in terms of the coefficients of ODE and one auxiliary function. This new criterion is used to obtain the general solutions for the first integral explicitly, providing a direct alternative procedure for constructing the first integrals and Sundman transforma- tions. The effectiveness of this approach is demonstrated by applying it to find the general solution for geodesics on surfaces of revolution of constant curvature in a unified manner.

A New Approach for Semi-Analytical Solution of Cross-plane Phonon Transport in Silicon–Diamond Thin Films

Abstract Transient analysis of phonon cross-plane transport across two consecutively placed thin films is considered, and a new approach is introduced to obtain the semi-analytical solution for the equation of phonon radiative transport. The orthogonality properties of trigonometric functions are used in the mathematical analysis. Silicon and diamond thin films are used to resemble the consecutively placed thin films. The films are thermally disturbed from its edges to initiate the phonon transport, and thermal boundary resistance is introduced at the films interface. Equivalent equilibrium temperature is incorporated to quantify the phonon intensity distribution in the films. It is found that the results of the analytical solution agree well with their counterparts obtained from the numerical simulations. Phonon intensity at the film edges and interface reduces significantly due to boundary scattering. The analytical solution captures phonon scattering at boundaries and interface correctly, and provides considerable simplification of the numerical treatment of the equation for phonon radiative transport. It also reduces significantly the numerical efforts required for solving the transient phonon radiative transport equation pertinent to the cross-plan transport across the thin films in terms of program size and run-time.

An alternative proof of Lie’s linearization theorem using a new λ-symmetry criterion

An alternative proof of Lie’s approach for linearization of scalar second order ODEs is derived using the relationship between λ-symmetries and first integrals. This relation further leads to a new λ-symmetry linearization criteria for second order ODEs which provides a new approach for constructing the linearization transformations with lower complexity. The effectiveness of the approach is illustrated by obtaining the local linearization transformations for the linearizable nonlinear ODEs of the form y + F1(x, y)y ′ + F (x, y) = 0. Examples of linearizing nonlinear ODEs which are quadratic or cubic in the first derivative are also presented.

Equilibrium Laser Pulse Heating and Thermal Stress Analysis

When the heating duration becomes greater than the thermalization time of the substrate material, equilibrium heating takes place in the laser irradiated region. In this case, the classical Fourier heating law governs the energy transport. Although the heating process is complicated, some useful assumptions enable to obtain the closed form solution for temperature and stress fields. Since the analytical solution provides the functional relation between the dependent variable and the independent parameters, it provides better physical insight into the heating problem than that of the numerical analysis. In this chapter, equilibrium heating of solid surfaces heated by a laser beam is considered. The closed form solution for the resulting temperature and stress fields are presented for various heating situations. The study also covers the phase change taking place at the irradiated region during the laser treatment process.

Analytical Treatment of Hyperbolic Equations for Stress Analysis

Laser ultra-short pulse heating of metallic surfaces causes the hyperbolic behavior of energy transport in the heated region. The consideration of the parabolic nature of the non-equilibrium heating situation fails to formulate the correct heating process. Although heating duration is ultra-short, material response to the heating pulse is not limited to only heat transfer and the mechanical response of the heated surface also becomes important. Consequently, mechanical response of the surface under ultra-short thermal loading becomes critical in terms of the generation of the high stress levels. In this chapter, hyperbolic behavior of heat transfer is introduced in the laser heated region. The closed for solutions for the temperature and stress fields are obtained for various heating situations. Two-dimensional effect of heating on temperature rise is also considered for nano-scale applications.

Analytical Solution of Cattaneo and Thermal Stress Equations

Laser short pulse heating of metallic surfaces initiates non-equilibrium energy transport in the irradiated region. In this case, thermal separation of electron and lattice sub-systems takes place. The thermal communication of these sub-systems occurs through the collisional process and the electrons transfer some of their excess energy during this process. Although electron temperature attains significantly high values due to the energy gain from the irradiated field through absorption, lattice site temperature remains low. Since the heated region is limited within a small volume, temperature gradients remain high across the irradiated region despite the attainment of low temperature field. Consequently, high temperature gradients cause the development of high thermal stress field in the small region. This limits the practical applications of the laser treatment process at microscopic scales. In this chapter, heat transfer at micro-scale is formulated and temperature field is presented analytically. The closed for solutions for the temperature and stress fields are obtained for various heating situations.