Derivation of the Finite-Difference Equations

In my final semester, my course load included a graduate course that had two modules: astronomical instrumentation and numerical modeling. The latter focused on developing the equations of motion of geophysical fluid dynamics (See Research in Magnetohydrodynamics). Such equations are then converted into an algorithm based on a specific type of numerical method of solving […]

Deriving the Bessel Function of the First Kind for Zeroth Order

NOTE: I verified the solution using the following text: Boyce, W. and DiPrima, R.¬†Elementary Differential Equations.¬† In this post, I shall be deriving the Bessel function of the first kind for the zeroth order Bessel differential equation. Bessel’s equation is encountered when solving differential equations in cylindrical coordinates and is of the form where describes […]

Consequences and some Elementary Theorems of the Ideal One-Fluid Magnetohydrodynamic Equations

SOURCE FOR CONTENT: Priest, E. Magnetohydrodynamics of the Sun, 2014. Cambridge University Press. Ch.2.; Davidson, P.A., 2001. An Introduction to Magnetohydrodynamics. Ch.4.¬† We have seen how to derive the induction equation from Maxwell’s equations assuming no charge and assuming that the plasma velocity is non-relativistic. Thus, we have the induction equation as being Many texts […]

Basic Equations of Ideal One-Fluid Magnetohydrodynamics: (Part V) The Energy Equations and Summary

SOURCE FOR CONTENT: Priest E., Magnetohydrodynamics of the Sun, 2014. Ch. 2. Cambridge University Press.   The final subset of equations deals with the energy equations. My undergraduate research did not take into account the thermodynamics of conducting fluid in order to keep the math relatively simple. However, in order to understand MHD one must […]