Generalized Bessel functions: Theory and their applications

This paper presents 2 new classes of the Bessel functions on a compact domain [0,T] as generalized‐tempered Bessel functions of the first‐ and second‐kind which are denoted by GTBFs‐1 and GTBFs‐2. Two special cases corresponding to the GTBFs‐1 and GTBFs‐2 are considered. We first prove that these functions are as the solutions of 2 linear differential operators and then show that these operators are self‐adjoint on suitable domains. Some interesting properties of these sets of functions such as orthogonality, completeness, fractional derivatives and integrals, recursive relations, asymptotic formulas, and so on are proved in detail. Finally, these functions are performed to approximate some functions and also to solve 3 practical differential equations of fractionalorders.     

On numerical improvement of Gauss–Legendre quadrature rules

Among all integration rules with n points, it is well-known that n-point Gauss–Legendre quadrature rule∫₋₁¹f(x) dx∑_i=1ⁿw_if(x_i)has the highest possible precision degree and is analytically exact for polynomials of degree at most 2n−1, where nodes x_i are zeros of Legendre polynomial P_n(x), and w_i's are corresponding weights.In this paper we are going to estimate numerical values of nodes x_i and weights w_i so that the absolute error of introduced quadrature rule is less than a preassigned tolerance ε₀, say ε₀=10⁻⁸, for monomial functionsf(x)=x^j, j=0,1,…,2n+1.(Two monomials more than precision degree of Gauss–Legendre quadrature rules.) We also consider some conditions under which the new rules act, numerically, more accurate than the corresponding Gauss–Legendre rules. Some examples are given to show the numerical superiority of presented rules.     

The general Jacobi matrix method for solving some nonlinear ordinary differential equations

In this paper, we obtain the approximate solutions for some nonlinear ordinary differential equations by using the general Jacobi matrix method. Explicit formulae which express the Jacobi expansion coefficients for the powers of derivatives and moments of any differentiable function in terms of the original expansion coefficients of the function itself are given in the matrix form. Three test problems are discussed to illustrate the efficiency of the proposed method.     

A numerical method based on extended Raviart–Thomas (ER‐T) mixed finite element method for solving damped Boussinesq equation

In this paper, we present the approximate solution of damped Boussinesq equation using extended Raviart–Thomas mixed finite element method. In this method, the numerical solution of this equation is obtained using triangular meshes. Also, for discretization in time direction, we use an implicit finite difference scheme. In addition, error estimation and stability analysis of both methods are shown. Finally, some numerical examples are considered to confirm the theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.     

Two families of scaled three-term conjugate gradient methods with sufficient descent property for nonconvex optimization

Numerical Algorithms - In this paper, we present two families of modified three-term conjugate gradient methods for solving unconstrained large-scale smooth optimization problems. We show that our...     

The weighted (0,1,…,m−2,m)-interpolation technique based on the roots of the classical orthogonal polynomials and application in deriving new quadrature rules

We investigate the problem of weighted (0,1,…,m?2,m)-interpolation (m≧2, ${m\in \mathbb{N}}$ ) on the roots of the classical orthogonal polynomials. The necessary and sufficient conditions for the existence and uniqueness of this problem are established. Meanwhile, the explicit representation (characterization) of the weighted (0,1,…,m?2,m)-interpolation is given. As an application we obtain a Birkhoff type quadrature formula which is exact for the polynomials of degree at most mn. Also we investigate the error function of the new (0,1,…,m?2,m)-Birkhoff type weighted quadrature formulae using Peano kernel theorem. Some numerical examples are given to support the results.     

A new approach to improve the order of approximation of the Bernstein operators: theory and applications

This paper presents a new approach to improve the order of approximation of the Bernstein operators. Three new operators of the Bernstein-type with the degree of approximations one, two, and three are obtained. Also, some theoretical results concerning the rate of convergence of the new operators are proved. Finally, some applications of the obtained operators such as approximation of functions and some new quadrature rules are introduced and the theoretical results are verified numerically.     

The convergence and stability analysis of the Jacobi collocation method for solving nonlinear fractional differential equations with integral boundary conditions

In this paper, we apply the Jacobi collocation method for solving nonlinear fractional differential equations with integral boundary conditions. Due to existence of integral boundary conditions, after reformulation of this equation in the integral form, the method is proposed for solving the obtained integral equation. Also, the convergence and stability analysis of the proposed method are studied in two main theorems. Furthermore, the optimum degree of convergence in the L₂ norm is obtained for this method. Furthermore, some numerical examples are presented in order to illustrate the performance of the presented method. Finally, an application of the model in control theory is introduced. Copyright © 2015 John Wiley & Sons, Ltd.     

PROSIGN: a method for protein secondary structure assignment based on three-dimensional coordinates of consecutive C(alpha) atoms

The automatic assignment of secondary structure from three-dimensional atomic coordinates of proteins is an essential step for the analysis and modeling of protein structures. So different methods based on different criteria have been designed to perform this task. We introduce a new method for protein secondary structure assignment based solely on C(alpha) coordinates. We introduce four certain relations between C(alpha) three-dimensional coordinates of consecutive residues, each of which applies to one of the four regular secondary structure categories: alpha-helix, 3(10)-helix, pi-helix and beta-strand. In our approach, the deviation of the C(alpha) coordinates of each residue from each relation is calculated. Based on these deviation values, secondary structures are assigned to all residues of a protein. We show that our method agrees well with popular methods as DSSP, STRIDE and assignments in PDB files. It is shown that our method gives more information about helix geometry leading to more accurate secondary structure assignment.