A Numerical Method for Solving Ill-Conditioned Equation Systems Arising from Radial Basis Functions

Continuously differentiable radial basis functions (C∞-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are full and can become very ill- conditioned. Similarly, the Hilbert and Vandermonde have full matrices and become ill-conditioned. The difference between a coefficient matrix generated by C∞-RBFs for partial differential or integral equations and Hilbert and Vandermonde systems is that C∞-RBFs are very sensitive to small changes in the adjustable parameters. These parameters affect the condition number and solution accuracy. The error terrain has many local and global maxima and minima. To find stable and accurate numerical solutions for full linear equation systems, this study proposes a hybrid combination of block Gaussian elimination (BGE) combined with arbitrary precision arithmetic (APA) to minimize the accumulation of rounding errors. In the future, this algorithm can execute faster using preconditioners and implemented on massively parallel computers.

A Technique for Estimation of Box Dimension about Fractional Integral

This paper discusses further the roughness of Riemann-Liouville fractional integral on an arbitrary fractal continuous functions that follows Rfs. [1]. A novel method is used to reach a similar result for an arbitrary fractal function , where is the Riemann-Liouville fractional integral. Furthermore, a general resultis arrived at for 1-dimensional fractal functions such as with unbounded variation and(or) infinite lengths, which can infer all previous studies such as [2] [3]. This paper’s estimation reveals that the fractional integral does not increase the fractal dimension of f(x), i.e. fractional integration does not increase at least the fractal roughness. And the result has partly answered the fractal calculus conjecture and completely answered this conjecture for all 1-dimensional fractal function (Xiao has not answered). It is significant with a comparison to the past researches that the box dimension connection between a fractal function and its Riemann-Liouville integral has been carried out only for Weierstrass type and Besicovitch type functions, and at most Hlder continuous. Here the proof technique for Riemann-Liouville fractional integral is possibly of methodology to other fractional integrals.

An Iterative Relaxation Approach to the Solution of the Hamilton-Jacobi-Bellman-Isaacs Equation in Nonlinear Optimal Control

In this paper, we propose an iterative relaxation method for solving the Hamilton-Jacobi-Bellman-Isaacs equation(HJBIE) arising in deterministic optimal control of affine nonlinear systems. Local convergence of the method is established under fairly mild assumptions, and examples are solved to demonstrate the effectiveness of the method. An extension of the approach to Lyapunov equations is also discussed. The preliminary results presented are promising, and it is hoped that the approach will ultimately develop into an efficient computational tool for solving the HJBIEs.

On the (<i>r</i>,<i>φ</i>,<i>ψ</i>)-Suzuki Contraction for the Multi-Valued Mappings

In this work we try to give a new contraction type in multi-valued mapping on complete metric spaces. We prove the existence of fixed point for (r,φ,ψ)-Suzuki contraction in such spaces. Around our paper, the function ψ is absolutely continuous, and in this case, the contraction proposed by as has a fixed point.

Intuitionistic Fuzzy α-Generalized Closed Sets in Terms of Minimal Structure Spaces

In this paper, we introduce the notion of intuitionistic fuzzy α-generalized closed sets in intuitionistic fuzzy minimal structure spaces and investigate some of their properties. Further, we introduce and study the concept of intuitionistic fuzzy α-generalized minimal continuous functions.

Numerical solution for singular differential equations using Haar wavelet

The aim of this paper is to obtain the numerical solution of singular ordinary differential equations using the Haar-wavelet approach.The proposed method is mathematically simple and provides highly accurate solutions.In this method,we derive the Haar operational matrix using Haar function.Haar operational matrix is a basic tool and applied in system analysis to evaluate the numerical solution of differential equations.The convergence of the proposed method is discussed through its error analysis.To illustrate the efficiency of this method,solutions of four singular differential equations are obtained.

Computational simulation methods for fiber reinforced composites

Trefftz-finite element method(Trefftz-FEM),adaptive cross approximation BEM(ACA BEM)and continuous source function method(CSFM)are used for the simulation of composites reinforced by short fibers(CRSF)with the aim of showing the possibilities of reducing the problem of complicated and important interactions in such composite materials.