If the particle can have an energy E < 0, it will be localized in and near the potential well, with a wave function that decays to zero as r increases to values greater than a. A simple case of this problem was one of our examples of an eigenvalue problem (Example 8.3.3), but ...
Bessel and Modified Bessel ODEs Description Examples Description The general form of the Bessel ODE is given by the following: Bessel_ode := x^2*diff(y(x),x,x)+x*diff(y(x),x)+(x^2-n^2)*y(x); The general form of the modified Bessel ODE is given by the...
But, often a real nonlinear FDE has not the exact or analytical solution and must be solved numerically. Therefore, we aim to introduce a new numerical algorithm based on generalized Bessel function of the first kind (GBF), spectral methods and Newton–Krylov subspace method to solve nonlinear ...
functionhavebeen presented. Chapter twoLevinmethodfor ff(x)J=(rx)dx hasbeendiscussed andtheerror analyze andsomenumerical examples havebeenshowed, Chapter three twokinds ofmethodsto solve problem like f缸)厶(唧D))凼has beendiscussed.OneisLevin ...
(5) is an operator equation of the first kind which cannot be solved directly, since from Lemma 3, the trace operator AN is compact, and we do not know if the function f is in the range AN(ℂ2N+1) of AN. Therefore, we will solve the ill-posed operator equation (5) by a ...
The above two examples lead to the following remarks: Nevertheless, we know that v¯n can be non-zero for odd n when the collided ions are not spherical or have different sizes.9 The delta function approximation for p(v2,x , v2,y) is not compatible with the experimental observation. ...
Bessel and Modified Bessel ODEs Description Examples Description The general form of the Bessel ODE is given by the following: Bessel_ode := x^2*diff(y(x),x,x)+x*diff(y(x),x)+(x^2-n^2)*y(x); The general form of the modified Bessel ODE is given by the...
Using the continuity results, these transforms are extended as an adjoint to the corresponding dual space of distributions. The examples of Bessel wavelet transform of a polynomial function and a distribution are given. The Schrdinger equation is solved using the Bessel wavelet transform....
Also, an error problem is constructed by the residual function, and it is solved by the presented method. Thus, the error function is estimated, and the approximate solutions are improved. Finally, numerical examples are given to show the applicability of the method, and also, our results are...
Hence, the Bessel function of the first kind (of order ν≠ an integer) is (2.168) The general solution of Eq. (2.156) for ν≠ an integer and for all x≠0 is (2.169) However, for integer ν=n, (2.170)J−n(x)=(−1)nJn(x)(n=1,2,...), and therefore J−n(x) and...