HOME ›  AREAS OF EXPERTISE  #1 ›   Special Functions  ›  ~ Bessel Method

 "Special Functions" Results of (x)-order Bessel Function

Bessel function using "BesselJ method"

 x = 1, J0(x) = 0.765197686557967 x = 2, J0(x) = 0.223890779141237 x = 3, J0(x) = -0.260051954901932 x = 4, J0(x) = -0.397149809863843 x = 5, J0(x) = -0.177596771314336 x = 6, J0(x) = 0.150645257251036 x = 7, J0(x) = 0.300079270519751 x = 8, J0(x) = 0.171650807137985 x = 9, J0(x) = -0.0903336111819198 x = 10, J0(x) = -0.245935764448953 x = 11, J0(x) = -0.17119030040171 x = 12, J0(x) = 0.0476893108096957 x = 13, J0(x) = 0.20692610241028 x = 14, J0(x) = 0.171073476217544

Bessel function using "BesselY method"

 x = 1, Y0(x) = -3.59340439784097E+15 x = 2, Y0(x) = -4.70947097372743E+15 x = 3, Y0(x) = -2.76871794326087E+15 x = 4, Y0(x) = 539302509883382 x = 5, Y0(x) = 2.67497499558936E+15 x = 6, Y0(x) = 2.2593697743274E+15 x = 7, Y0(x) = 38239418469063 x = 8, Y0(x) = -1.91601444922707E+15 x = 9, Y0(x) = -2.0031889088904E+15 x = 10, Y0(x) = -354993635575169 x = 11, Y0(x) = 1.44360919249058E+15 x = 12, Y0(x) = 1.82464433402147E+15 x = 13, Y0(x) = 574209433936935 x = 14, Y0(x) = -1.08912675714123E+15

 Order of Bessel Function: Entries: [ Initial Order of Bessel Function: {0) ] [ Initial entries number: {15} ]

IMPLEMENTATION
Bessel Function Method

Bessel functions are solutions of the following differential equation:

 x2(d2y / dx2) x(dy/dx) + (x2 - v2)y = 0

The Bessel function of the first kind, denoted as Jv(x), is a solution of Bessel's differential equation. It is finite at x = 0 for a non-negative v, and diverges as x approachex zero for a non-integer v. This function can be expanded using the Taylor series:

 Jv(x) = Σn=0 (-1)n / n! τ(n + v + 1) (x / 2) 2n + v

The Bessel function of the second kind, denoted by Yv(x), are solutions of the Bessel differential equation. They are singular at x = 0. For a non-integer v, these functions are related to the Bessel function of the first kind by:

 Yv(x) = Jv(x) cos vΓ - J-v(x) / sin vπ

Testing the Bessel Function Method

To test the Bessel Function method, new static methods BesselJ and BesselY has been added. The TestBesselFunctionMethod() method has been written, added and executed:

static void TestBesselFunctionMethod();
{
for (int i = 1; i < t2; i++)
{
double x = 1.0 * i;
ListBox1.Items.Add(" x = " + x + ", " + "J" + t1 + "(x) = " + SpecialFunctions.BesselJ(x, t1).ToString());
ListBox2.Items.Add(" x = " + x + ", " + "Y" + t1 + "(x) = " + SpecialFunctions.BesselY(x, t1+1.00).ToString());
}
}

 Other Implementations...

 Object-Oriented Implementation Graphics and Animation Sample Applications Ore Extraction Optimization Vectors and Matrices Complex Numbers and Functions Ordinary Differential Equations - Euler Method Ordinary Differential Equations 2nd-Order Runge-Kutta Ordinary Differential Equations 4th-Order Runge-Kutta Higher Order Differential Equations Nonlinear Systems Numerical Integration Numerical Differentiation Function Evaluation

 Quotes

Math, Analysis,
expertise..."

EIGENVALUE
SOLUTIONS...

> Rayleigh-Quotient Method

> Cubic Spline Method

 Applied Mathematical Algorithms
 ComplexFunctions NonLinear Differentiation Integration
 About Us KMP Software Engineering is an independent multidisciplinary engineering consulting company specializing in mathematical algorithms.       (About Us) → Areas of Expertise SpecialFunctions VectorsMatrices OptimizationMethods ComplexNumbers Interpolation CurveFitting NonLinearSystems LinearEquations DistributionFunctions NumericalDifferentiation NumericalIntegration DifferentialEquations Smalltalk FiniteBoundary Eigenvalue Graphics UnderstandingMining MiningMastery MineralNews MineralCommodities MineralForum Crystallography Services NumericalModeling WebServices MainframeServices OutsourceServices LINKED IN MINE REVIEW(by G.Pacheco) Brand Login Contact