Introduction
The main objective of this chapter is familiarization with a variety of numerical methods that are essential for solving advanced problems of applied physics and engineering. With the help of suitable examples, basic skills on appropriately using these methods for various applications in physics are provided.
The chapter focuses on the following special second order differential equations, which are known to have standard functional form and/or analytical solutions.
• Bessel's equation (Section 6.2)
• Legendre's equation (Section 6.3)
• Laguerre's equation (Section 6.4)
• Hermite's equation (Section 6.5)
The solutions of these equations are referred to as ‘special functions’, which are significantly different from standard functions like sine/cosine, exponential and logarithmic functions. This chapter also describes the use of quadrature methods of integration for calculating improper integrals, which are either infinite in the interval of integration, or the interval of integration has an infinite bound. The quadrature methods discussed in this chapter are as follows:
• Gauss–Legendre (Section 6.6.1)
• Gauss–Laguerre (Section 6.6.2)
• Gauss–Hermite (Section 6.6.3)
The chapter has been written in a manner so as to develop the necessary skills of the reader to evaluate certain integrals that are generally not discussed in introductory physics classes because they involve advanced calculations.
Bessel Function of the First Kind
Bessel functions have several applications in physics. They arise while solving Laplace's and Helmholtz equations in spherical and cylindrical coordinates. The functions are also useful while solving problems based on electromagnetic wave propagation and Schrödinger's equation.
The general features of the Bessel function are as follows.
1. Bessel functions (order n) of the first kind (Jn(x)) are the solutions (y(x)) of the differential equation given in Eqn. 6.1.
A second order differential equation can be written in the form
For solving Eqn. 6.1 with the finite difference method, it is necessary to first define the functions f(x), g(x), and r(x)in the following manner.
The function for the finite difference method has already been explained in detail in Chapter 4. This function can be written in an executable file, ‘differentiation.sci’ (for example) and can be loaded using the following Scilab command.
Figure 6.1 shows the zero order Bessel function of the first kind. It has been generated using the following Scilab program.
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