Persamaan Legendere dan Fungs Bessel.pptx

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1. Solusi Persamaan Differensial Persamaan legendre Persamaan bessel

2. METODE FROBENIUS Metode untuk memecahkan solusi dari persamaan differensial dengan koefisien variable: p(x) dan q(x) adalah koefisien variable (variable co- effecients ).

3. Solusi Persamaan Differensial 1. Persamaan legendre 2. Persamaan bessel

4. Solusi Persamaan Differensial Solusinya dinyatakan sebagai berikut . sehingga ,      

5. Penggunaan Dalam Fisika Hantaran Panas ( Termodinamika dan Fisika Statistik ) Potensial dan Medan Listrik( Teori Elektromagnetik )

6. Persamaan Legendre Solusinya dengan Metode Frobenius : Maka :

7. Polinomial Legendre   Jika dan dipilih sehingga y=1 ketika x=1, maka penyelesaian yang diperolejdisebut Polinomial Legendre  

8. Polinomial Legendre dengan Formula Rodriguez   Formula Rodriguez adalah rumus yang dapat dipakai untuk mencari polynomial legendre ke - .  

9. Polinomial Legendre dengan Fungsi Pembangkit   Fungsi Pembangkit Polinomial Legendre: Maka :

10. Polinomial Legendre dengan Fungsi Pembangkit   Dari Fungsi Pembangkit Polinomial Legendre, akan didapatkan Hubungan Rekursi untuk mendapatkan Polinomial Legendre:

11. Ortogonalitas Polinomial Legendre

12. Deret Legendre

13. Fungsi Legendre Terasosiasi

14. Persamaan Bessel Solusinya dengan Metode Frobenius : Maka :

15. Solutions about Ordinary Points (cont’d.) If is an ordinary point of the DE, we can always find two linearly independent solutions in the form of a power series centered at x A series solution converges at least on some interval defined by where R is the distance from x to the closest singular point This R is the lower bound for radius of convergence

16. Solutions about Singular Points Consider the linear second-order DE Divide by to put into standard form Point x is a regular singular point of the DE if both and are analytic at x A singular point that is not regular is an irregular singular point of the equation

17. Solutions about Singular Points (cont’d.) To solve a DE about a regular singular point, we employ Frobenius’ Theorem If is a regular singular point of the standard DE, there exists at least one nonzero solution of the form where r is a constant, and the series converges at least on some interval

18. Solutions about Singular Points (cont’d.) After substituting into a DE and simplifying, the indicial equation is a quadratic equation in r that results from equating the total coefficient of the lowest power of x to zero The indicial roots are the solutions to the quadratic equation The roots are then substituted into a recurrence relation

19. Solutions about Singular Points (cont’d.) Suppose that is a regular singular point of a DE and indicial roots are r 1 and r 2 Case 1: r 1 and r 2 are distinct and do not differ by an integer, then there are two linearly independent solutions

20. Solutions about Singular Points (cont’d.) Suppose that is a regular singular point of a DE and indicial roots are r 1 and r 2 Case 2: r 1 – r 2 = N where N is a positive integer, then there are two linearly independent solutions Case 3: r 1 = r 2 , then there are two linearly independent solutions

21. Special Functions The following DEs occur frequently in advanced mathematics, physics, and engineering Bessel’s equation of order v , solutions are Bessel functions Legendre’s equation of order n , solutions are Legendre functions

22. Special Functions (cont’d.) There are various formulations of the solutions to Bessel’s equation Bessel’s functions of the first kind Bessel’s functions of the second kind Bessel’s functions of half-integral order Spherical Bessel functions of the first and second kind

23. Special Functions (cont’d.) Legendre polynomials , , are specific n th-degree polynomial solutions The first few Legendre polynomials are