Abstract

Partial differential equations (PDEs) are extremely important in both mathematics and physics. This chapter provides an introduction to some of the simplest and most important PDEs in both disciplines, and techniques for their solution. The chapter focuses on three equations—the heat equation, the wave equation, and Laplace's equation. Following the nomenclature of the geometrical figures, if B 2 – 4AC 2 – 4AC=0 the equation is elliptic; and if B 2 – 4AC>0 the equation is hyperbolic. Thus, the heat equation is the prototypical parabolic equation, the wave equation is the prototypical wave equation, and Laplace's equation is the prototypical elliptical equation. It is shown that for the heat equation the initial conditions diffuse in time, whereas in the wave equation initial conditions are propagated, changing position but not shape.

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.