In this Paper, we have proposed a new weighted residual method known as orthogonal collocation-based on mixed interpolation (OCMI). Mixed interpolation uses the classical polynomial approximation with two correction t...In this Paper, we have proposed a new weighted residual method known as orthogonal collocation-based on mixed interpolation (OCMI). Mixed interpolation uses the classical polynomial approximation with two correction terms given in the form of sine and cosine function. By these correction terms, we can control the error in the solution. We have applied this approach to a non-linear boundary value problem (BVP) in ODE which governs the electrohydrodynamic flow in a cylindrical conduit. The solution profiles shown in the figures are in good agreement with the work of Paullet (1999) and Ghasemi et al. (2014). Our solution is monotonic decreasing and satisfies , where, α governs the strength of non-linearity and for large values of α solutions are . The residual errors are given in Table 1 and Table 2 which are significantly small. Comparison of residual errors between our proposed method, Least square method and Homotopy analysis method is also given and shown via the Table 3 where as the profiles of the residual error are depicted in Figures 4-8. Table and graphs show that efficiency of the proposed method. The error bound and its L2-norm with relevant theorems for mixed interpolation are also given.展开更多
By combining the classical appropriate functions “1, x, x 2” with the method of multiplier enlargement, this paper establishes a theorem to approximate any unbounded continuous functions with modified positive...By combining the classical appropriate functions “1, x, x 2” with the method of multiplier enlargement, this paper establishes a theorem to approximate any unbounded continuous functions with modified positive linear operators. As an example, Hermite Fejér interpolation polynomial operators are analysed and studied, and a general conclusion is obtained.展开更多
In this paper,an optimality condition for nonlinear programming problems with box constraints is given by using linear transformation and Lagrange interpolating polynomials.Based on this condition,two new local optim...In this paper,an optimality condition for nonlinear programming problems with box constraints is given by using linear transformation and Lagrange interpolating polynomials.Based on this condition,two new local optimization methods are developed.The solution points obtained by the new local optimization methods can improve the Karush–Kuhn–Tucker(KKT)points in general.Two global optimization methods then are proposed by combining the two new local optimization methods with a filled function method.Some numerical examples are reported to show the effectiveness of the proposed methods.展开更多
When one solves differential equations by a spectral method,it is often convenient to shift from Chebyshev polynomials Tn(x) with coefficients anto modified basis functions that incorporate the boundary conditions.For...When one solves differential equations by a spectral method,it is often convenient to shift from Chebyshev polynomials Tn(x) with coefficients anto modified basis functions that incorporate the boundary conditions.For homogeneous Dirichlet boundary conditions,u(±1)=0,popular choices include the "Chebyshev difference basis" ζn(x)≡Tn+2(x)-Tn(x) with coefficients here denoted by bnand the "quadratic factor basis" Qn(x)≡(1-x2)Tn(x) with coefficients cn.If u(x) is weakly singular at the boundary,then the coefficients andecrease proportionally to O(A(n)/nκ) for some positive constant κ,where A(n) is a logarithm or a constant.We prove that the Chebyshev difference coefficients bndecrease more slowly by a factor of 1/n while the quadratic factor coefficients cndecrease more slowly still as O(A(n)/nκ-2).The error for the unconstrained Chebyshev series,truncated at degree n=N,is O(|A(N)|/Nκ) in the interior,but is worse by one power of N in narrow boundary layers near each of the endpoints.Despite having nearly identical error norms in interpolation,the error in the Chebyshev basis is concentrated in boundary layers near both endpoints,whereas the error in the quadratic factor and difference basis sets is nearly uniformly oscillating over the entire interval in x.Meanwhile,for Chebyshev polynomials,the values of their derivatives at the endpoints are O(n2),but only O(n) for the difference basis.Furthermore,we give the asymptotic coefficients and rigorous error estimates of the approximations in these three bases,solved by the least squares method.We also find an interesting fact that on the face of it,the aliasing error is regarded as a bad thing;actually,the error norm associated with the downward curving spectral coefficients decreases even faster than the error norm of infinite truncation.But the premise is under the same basis,and when involving different bases,it may not be established yet.展开更多
文摘In this Paper, we have proposed a new weighted residual method known as orthogonal collocation-based on mixed interpolation (OCMI). Mixed interpolation uses the classical polynomial approximation with two correction terms given in the form of sine and cosine function. By these correction terms, we can control the error in the solution. We have applied this approach to a non-linear boundary value problem (BVP) in ODE which governs the electrohydrodynamic flow in a cylindrical conduit. The solution profiles shown in the figures are in good agreement with the work of Paullet (1999) and Ghasemi et al. (2014). Our solution is monotonic decreasing and satisfies , where, α governs the strength of non-linearity and for large values of α solutions are . The residual errors are given in Table 1 and Table 2 which are significantly small. Comparison of residual errors between our proposed method, Least square method and Homotopy analysis method is also given and shown via the Table 3 where as the profiles of the residual error are depicted in Figures 4-8. Table and graphs show that efficiency of the proposed method. The error bound and its L2-norm with relevant theorems for mixed interpolation are also given.
文摘By combining the classical appropriate functions “1, x, x 2” with the method of multiplier enlargement, this paper establishes a theorem to approximate any unbounded continuous functions with modified positive linear operators. As an example, Hermite Fejér interpolation polynomial operators are analysed and studied, and a general conclusion is obtained.
基金the National Natural Science Foundation of China(No.11471062).
文摘In this paper,an optimality condition for nonlinear programming problems with box constraints is given by using linear transformation and Lagrange interpolating polynomials.Based on this condition,two new local optimization methods are developed.The solution points obtained by the new local optimization methods can improve the Karush–Kuhn–Tucker(KKT)points in general.Two global optimization methods then are proposed by combining the two new local optimization methods with a filled function method.Some numerical examples are reported to show the effectiveness of the proposed methods.
基金supported by National Science Foundation of USA (Grant No. DMS1521158)National Natural Science Foundation of China (Grant No. 12101229)+1 种基金the Hunan Provincial Natural Science Foundation of China (Grant No. 2021JJ40331)the Chinese Scholarship Council (Grant Nos. 201606060017 and 202106720024)。
文摘When one solves differential equations by a spectral method,it is often convenient to shift from Chebyshev polynomials Tn(x) with coefficients anto modified basis functions that incorporate the boundary conditions.For homogeneous Dirichlet boundary conditions,u(±1)=0,popular choices include the "Chebyshev difference basis" ζn(x)≡Tn+2(x)-Tn(x) with coefficients here denoted by bnand the "quadratic factor basis" Qn(x)≡(1-x2)Tn(x) with coefficients cn.If u(x) is weakly singular at the boundary,then the coefficients andecrease proportionally to O(A(n)/nκ) for some positive constant κ,where A(n) is a logarithm or a constant.We prove that the Chebyshev difference coefficients bndecrease more slowly by a factor of 1/n while the quadratic factor coefficients cndecrease more slowly still as O(A(n)/nκ-2).The error for the unconstrained Chebyshev series,truncated at degree n=N,is O(|A(N)|/Nκ) in the interior,but is worse by one power of N in narrow boundary layers near each of the endpoints.Despite having nearly identical error norms in interpolation,the error in the Chebyshev basis is concentrated in boundary layers near both endpoints,whereas the error in the quadratic factor and difference basis sets is nearly uniformly oscillating over the entire interval in x.Meanwhile,for Chebyshev polynomials,the values of their derivatives at the endpoints are O(n2),but only O(n) for the difference basis.Furthermore,we give the asymptotic coefficients and rigorous error estimates of the approximations in these three bases,solved by the least squares method.We also find an interesting fact that on the face of it,the aliasing error is regarded as a bad thing;actually,the error norm associated with the downward curving spectral coefficients decreases even faster than the error norm of infinite truncation.But the premise is under the same basis,and when involving different bases,it may not be established yet.
