Introduction to Partial Differential Equations

This chapter provides an introduction to partial differential equations. It describes how the simplest partial differential equations arise and additionally provides ample examples. The chapter gives the definition of the Laplace operator. It also gives the expression of the field action approach. The chapter provides the Fourier law that states that for an isotropic medium, the flow of heat is in the direction of decreasing temperature and is proportional to the rate of this decrease. It further presents the case of diffusion to which Fourier's law is adapted. The chapter illustrates hyperbolic, elliptic, and parabolic differential equations, and focuses on their differences. It additionally illustrates Green's theorem and Green's function for linear and elliptic differential equations. The chapter provides the definition of a unit source and of the principle solution. The analytic character of the solution of an elliptical differential equation is discussed in the chapter. The definition of Green's function for self-adjoint differential equation is also given in the chapter. The chapter further illustrates Riemann's integration of the hyperbolic differential equation. It focuses on Green's theorem in heat conduction and explains that heat expands with infinite velocity.