Partial Differential Equations - Lecture 1 Notes

SMA 2371: PARTIAL DIFFERENTIAL EQUATIONS
⃝Francis
c O. Ochieng


Department of Pure and Applied Mathematics
Jomo Kenyatta University of Agriculture and Technology


Course content
• Surfaces and curves in three dimensions. Simultaneous differential equations of the first order.
dx dy dz
Methods of solution of = = . Orthogonal trajectories of system of curves on a surface.
P Q R
• Linear partial differential equations of the first order.

• Partial differential equations of the second order: Laplace, Poisson, heat and wave equations.
Methods of the solution by separation of the variables for Cartesian, spherical polar and cylin-
drical polar coordinates, and by Laplace and Fourier transform.

• Applications to engineering.


References
[1] Elements of PDE by Ian N. Sneddon

[2] Advanced Engineering Mathematics (10th ed.) by Erwin Kreyszig

[3] Schaum’s Outline Series: Theory and problems of PDE


Lecture 1


1 Review of basic concepts
1.1 Partial differentials
Let u = u(x, y), where x = x(s, t) and y = y(s, t). The partial differential of u is defined by

∂u ∂u
∂u = ∂x + ∂y.
∂x ∂y
Therefore, the partial derivative of u with respect to s and t is defined by
∂u ∂u ∂x ∂u ∂y
= +
∂s ∂x ∂s ∂y ∂s
and
∂u ∂u ∂x ∂u ∂y
= + ,
∂t ∂x ∂t ∂y ∂t
respectively.

Example(s):




1