Applied Mathematics Commons^™

(R2067) Solutions Of Hyperbolic System Of Time Fractional Partial Differential Equations For Heat Propagation, Sagar Sankeshwari, Vinayak Kulkarni

Applications and Applied Mathematics: An International Journal (AAM)

Hyperbolic linear theory of heat propagation has been established in the framework of a Caputo time fractional order derivative. The solution of a system of integer and fractional order initial value problems is achieved by employing the Adomian decomposition approach. The obtained solution is in convergent infinite series form, demonstrating the method’s strengths in solving fractional differential equations. Moreover, the double Laplace transform method is employed to acquire the solution of a system of integer and fractional order boundary conditions in the Laplace domain. An inversion of double Laplace transforms has been achieved numerically by employing the Xiao algorithm in …

On Solutions Of First Order Pde With Two-Dimensional Dirac Delta Forcing Terms, Ian Robinson

Rose-Hulman Undergraduate Mathematics Journal

We provide solutions of a first order, linear partial differential equation of two variables where the nonhomogeneous term is a two-dimensional Dirac delta function. Our results are achieved by applying the unilateral Laplace Transform, solving the subsequently transformed PDE, and reverting back to the original space-time domain. A discussion of existence and uniqueness of solutions, a derivation of solutions of the PDE coupled with a boundary and initial condition, as well as a few worked examples are provided.

Homogenization In Perforated Domains And With Soft Inclusions, Brandon C. Russell

Brandon Russell

Homogenization In Perforated Domains And With Soft Inclusions, Brandon C. Russell

Theses and Dissertations--Mathematics

In this dissertation, we first provide a short introduction to qualitative homogenization of elliptic equations and systems. We collect relevant and known results regarding elliptic equations and systems with rapidly oscillating, periodic coefficients, which is the classical setting in homogenization of elliptic equations and systems. We extend several classical results to the so called case of perforated domains and consider materials reinforced with soft inclusions. We establish quantitative H¹-convergence rates in both settings, and as a result deduce large-scale Lipschitz estimates and Liouville-type estimates for solutions to elliptic systems with rapidly oscillating periodic bounded and measurable coefficients. Finally, …

Quasioptimality Of Some Spectral Mixed Methods, Jay Gopalakrishnan, Leszek Demkowicz

Mathematics and Statistics Faculty Publications and Presentations

In this paper, we construct a sequence of projectors into certain polynomial spaces satisfying a commuting diagram property with norm bounds independent of the polynomial degree. Using the projectors, we obtain quasioptimality of some spectralmixed methods, including the Raviart–Thomas method and mixed formulations of Maxwell equations. We also prove some discrete Friedrichs type inequalities involving curl.