Visualisation of Polynomials Used in Series Expansions
نویسندگان
چکیده
Boltzmann’s Equation describes a myriad of phenomena from gas and fluid flow to electrons in semiconductors and hence plays an essential role in todays physics. However, calculating a solution to this seven-dimensional partial differential equation is very difficult. The high dimensionality of the equation poses a problem for its solution as traversal mechanisms for these high dimensions are not generally available. We present spherical harmonics which serve as the basis for an alternative, direct solution method to Boltzmann’s equation. Here the solution is calculated by expanding the it into spherical harmonics and determining the corresponding coefficients. We visualise the spherical harmonics themselves, their changes due to the recursion relations, and compare their evolution.
منابع مشابه
The coefficients of differentiated expansions of double and triple Jacobi polynomials
Formulae expressing explicitly the coefficients of an expansion of double Jacobi polynomials which has been partially differentiated an arbitrary number of times with respect to its variables in terms of the coefficients of the original expansion are stated and proved. Extension to expansion of triple Jacobi polynomials is given. The results for the special cases of double and triple ultraspher...
متن کاملRecurrences and explicit formulae for the expansion and connection coefficients in series of the product of two classical discrete orthogonal polynomials
Suppose that for an arbitrary function $f(x,y)$ of two discrete variables, we have the formal expansions. [f(x,y)=sumlimits_{m,n=0}^{infty }a_{m,n},P_{m}(x)P_{n}(y),] $$ x^{m}P_{j}(x)=sumlimits_{n=0}^{2m}a_{m,,n}(j)P_{j+m-n}(x),$$ we find the coefficients $b_{i,j}^{(p,q,ell ,,r)}$ in the expansion $$ x^{ell }y^{r},nabla _{x}^{p}nabla _{y}^{q},f(x,y)=x^{ell }y^{r}f^{(p,q)}(x,y) =sumli...
متن کاملExpansions of Distributions in Term of Generalized Heat Polynomials and Their Appell Transforms
This paper is concerned with expansions of distributions in terms of the generalized heat polynomials and of their Appell transforms. Two different techniques are used to prove theorems concerning expansions of distributions. A theorem which provides an orthogonal series expansion of generalized functions is also established. It is shown that this theorem gives an inversion formula for a certai...
متن کاملRational series for multiple zeta and log gamma functions
We give series expansions for the Barnes multiple zeta functions in terms of rational functions whose numerators are complex-order Bernoulli polynomials, and whose denominators are linear. We also derive corresponding rational expansions for Dirichlet L-functions and multiple log gamma functions in terms of higher order Bernoulli polynomials. These expansions naturally express many of the well-...
متن کاملSome applications of the Hermite matrix polynomials series expansions 1
This paper deals with Hermite matrix polynomials expansions of some relevant matrix functions appearing in the solution of di erential systems. Properties of Hermite matrix polynomials such as the three terms recurrence formula permit an e cient computation of matrix functions avoiding important computational drawbacks of other well-known methods. Results are applied to compute accurate approxi...
متن کامل