![]() We can now apply Gauss Legendre quadrature formulas which are derived earlier for I to evaluate the integral I=1n2∫∫TH(X,Y)dXdY. ![]() Notice that integrating f(x, y) with respect to y is the inverse operation of taking the partial derivative of f(x, y) with respect to y. We have again shown that the use of affine transformation over each Ti and the use of linearity property of integrals lead to the result:I=∑i=1n×n∫∫Tif(x,y)dxdy=1n2∫∫TH(X,Y)dXdY,where H(X,Y)=∑i=1n×nf(xi(X,Y),yi(X,Y)) and x=xi(X,Y) and y=yi(X,Y) refer to affine transformations which map each Ti in (x,y) space into a standard triangular surface T in (X,Y) space. This process of going through two iterations of integrals is called double integration, and the last expression in Equation 3.1.1 is called a double integral. We then propose the discretisation of the standard triangular surface T into n2 right isosceles triangular surfaces Ti (i=1(1)n2) each of which has an area equal to 1/(2n2) units. Abstract The calculation method of double integral is as follows (1) Define the method (2) The double integral is calculated using rectangular coordinates (3) Double integrals are. We then apply the one dimensional Gauss Legendre quadrature rules in ξ and η variables to arrive at an efficient quadrature rule with new weight coefficients and new sampling points. The self-similar solution is of the second kind, and it satisfies boundary conditions corresponding to a nonzero constant spectrum (with all its derivative being zero) at $\omega=0$ and a power-law asymptotic $n(\omega) \to \omega^. This solution presumably corresponds to an asymptotic behavior of a spectrum evolving from a broad class of initial data, and it features a non-equilibrium finite-time condensation of the wave spectrum $n(\omega)$ at the zero frequency $\omega$. We study a self-similar solution of the kinetic equation describing weak wave turbulence in Bose-Einstein condensates. This procedures allow to achieve a solution with accuracy ≈4.7% which is realized for x*≈1.22. ![]() To solve this problem we develop a new high-precision algorithm based on Chebyshev approximations and double exponential formulas for evaluating the collision integral, as well as the iterative techniques for solving the integro-differential equation for the self-similar shape function. There are two options, which depend on how you want to use the PDF content: To show content from a PDF on a slide Take a picture of the part of a PDF that you. finding the value x* of the exponent x for which these two boundary conditions can be satisfied simultaneously. Finding it amounts to solving a nonlinear eigenvalue problem, i.e. The self-similar solution is of the second kind, and it satisfies boundary conditions corresponding to a nonzero constant spectrum (with all its derivative being zero) at ω=0 and a power-law asymptotic n(ω)→ω−x at ω→∞x∈R . This solution presumably corresponds to an asymptotic behavior of a spectrum evolving from a broad class of initial data, and it features a non-equilibrium finite-time condensation of the wave spectrum n(ω) at the zero frequency ω. ![]() We study a self-similar solution of the kinetic equation describing weak wave turbulence in Bose–Einstein condensates. ![]()
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