Load Diffusion and Absorption Problems From a Finite Fiber to Elastic Infinite Matrix

1994 ◽  
Vol 61 (3) ◽  
pp. 567-574 ◽  
Author(s):  
Ven-Gen Lee ◽  
Toshio Mura
Keyword(s):  
Integral Equations ◽  
Load Transfer ◽  
Approximate Solutions ◽  
Infinite Matrix ◽  
Algebraic Equations ◽  
Equivalent Inclusion Method ◽  
Equivalent Inclusion ◽  
Transfer Behavior ◽  
Ring Elements ◽  
Inhomogeneity Problem

The load transfer behavior of a finite fiber perfectly bonded to an infinite matrix of distinct elastic moduli is investigated in this paper. The fiber is subjected to the uniformly distributed loading applied at infinity or on one cross-section of the fiber. The stress disturbance due to the existing fiber is simulated by the equivalent inclusion method, which formulates the inhomogeneity problem to a system of integral equations. By dividing the fiber into finite numbers of ring elements with uniform distributed eigenstrains, the integral equations can be further reduced to a system of algebraic equations with coefficients expressed in terms of the integrals of Lipschitz-Hankel type. Numerical results are presented for resultant axial force for various fiber length and material properties. The limiting cases of the infinite and semi-infinite fibers are also compared with the exact and approximate solutions.

Load Transfer From a Finite Cylindrical Fiber Into an Elastic Half-Space

1994 ◽  
Vol 61 (4) ◽  
pp. 971-975 ◽  
Author(s):  
Ven-Gen Lee ◽  
Toshio Mura
Keyword(s):  
Integral Equations ◽  
Half Space ◽  
Load Transfer ◽  
Transfer Problem ◽  
Numerical Procedure ◽  
Elastic Half Space ◽  
Equivalent Condition ◽  
Algebraic Equations ◽  
Equivalent Inclusion Method ◽  
Cylindrical Fiber

Based on the equivalent inclusion method, the load transfer problem of a finite cylindrical fiber embedded in an elastic half-space of different elastic properties is presented. The equivalent condition of inhomogeneity and inclusion problems simulates the fiber to an inclusion with chosen eigenstrains, and the problem is formulated to a set of integral equations with the unknown strength of eigenstrains. A numerical procedure is developed using a discretizing scheme by which the set of integral equations is reduced to a system of algebraic equations.

Application of Bernoulli wavelet method to solve a system of fuzzy integral equations

2018 ◽  
Vol 9 (1-2) ◽  
pp. 16-27 ◽  
Author(s):  
Mohamed Abdel- Latif Ramadan ◽  
Mohamed R. Ali
Keyword(s):  
Numerical Method ◽  
Integral Equations ◽  
Bernoulli Polynomials ◽  
Approximate Solutions ◽  
Fredholm Integral Equations ◽  
Fuzzy Integral ◽  
Algebraic Equations ◽  
Wavelet Method ◽  
Numerical Examples ◽  
Bernoulli Wavelet

In this paper, an efficient numerical method to solve a system of linear fuzzy Fredholm integral equations of the second kind based on Bernoulli wavelet method (BWM) is proposed. Bernoulli wavelets have been generated by dilation and translation of Bernoulli polynomials. The aim of this paper is to apply Bernoulli wavelet method to obtain approximate solutions of a system of linear Fredholm fuzzy integral equations. First we introduce properties of Bernoulli wavelets and Bernoulli polynomials, then we used it to transform the integral equations to the system of algebraic equations. The error estimates of the proposed method is given and compared by solving some numerical examples.

scholarly journals Solving of Linear Volterra-Fredholm Integral Equations via Modification of Block Pulse Functions

2021 ◽  
Vol 17 (1) ◽  
pp. 33
Author(s):  
Ayyubi Ahmad
Keyword(s):  
Approximate Solution ◽  
Integral Equations ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Computational Method ◽  
Fredholm Integral Equations ◽  
Triangular System ◽  
Algebraic Equations ◽  
Block Pulse Functions ◽  
Linear Algebraic Equations

A computational method based on modification of block pulse functions is proposed for solving numerically the linear Volterra-Fredholm integral equations. We obtain integration operational matrix of modification of block pulse functions on interval [0,T). A modification of block pulse functions and their integration operational matrix can be reduced to a linear upper triangular system. Then, the problem under study is transformed to a system of linear algebraic equations which can be used to obtain an approximate solution of  linear Volterra-Fredholm integral equations. Furthermore, the rate of convergence is  O(h) and error analysis of the proposed method are investigated. The results show that the approximate solutions have a good of efficiency and accuracy.

Multiple ellipsoidal/elliptical inhomogeneities embedded in infinite matrix by equivalent inhomogeneous inclusion method

2021 ◽  
pp. 108128652110071
Author(s):  
Xiu-wei Yu ◽  
Zhong-wei Wang ◽  
Hao Wang
Keyword(s):  
Separation Distance ◽  
Stress And Strain ◽  
Infinite Matrix ◽  
The Finite Element Method ◽  
Stress Concentrations ◽  
Strain Fields ◽  
Equivalent Inclusion Method ◽  
Inclusion Method ◽  
Equivalent Inclusion ◽  
Small Separation

Traditional equivalent inclusion method provides unreliable predictions of the stress concentrations of two spherical inhomogeneities with small separation distance. This paper determines the stress and strain fields of multiple ellipsoidal/elliptical inhomogeneities by equivalent inhomogeneous inclusion method. Equivalent inhomogeneous inclusion method is an inverse of equivalent inclusion method and substitutes the subdomains of matrix with known strains by equivalent inhomogeneous inclusions. The stress and strain fields of multiple inhomogeneities are decomposed into the superposition of matrix under applied load and each solitary inhomogeneous inclusion with polynomial eigenstrains by the iteration of equivalent inhomogeneous inclusion method. Multiple circular and spherical inhomogeneities are respectively used as examples and examined by the finite element method. The stress concentrations of multiple inhomogeneities with small separation distances are well predicted by equivalent inhomogeneous inclusion method and the accuracies improve with the increase of eigenstrain orders. Equivalent inhomogeneous inclusion method gives more accurate stress predictions than equivalent inclusion method in the problem of two spherical inhomogeneities.

scholarly journals Approximate Solution and its Convergence Analysis for Hypersingular Integral Equations

2019 ◽  
Vol 69 (2) ◽  
pp. 173-178
Author(s):  
Vaishali Sharma ◽  
Amit Setia
Keyword(s):  
Approximate Solution ◽  
Integral Equations ◽  
Analytic Solution ◽  
Singular Integral Equations ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Hypersingular Integral ◽  
Hypersingular Integral Equations ◽  
Flight Vehicles ◽  
Theoretical Results

This paper propose a residual based Galerkin method with Legendre polynomial as a basis functions to find the approximate solution of hypersingular integral equations. These equations   occur quite naturally in the field of aeronautics  such as  problem of aerodynamics of flight vehicles  and during mathematical modeling  of  vortex  wakes  behind  aircraft.  The analytic solution of these kind of equations is known only for a particular case ( m(x,t) =0 in Eqn (1)).  Also,   in these  singular integral equations which occur during the formulation of many boundary value problems, the known function  m(x,t) in (Eqn (1)) is not always   zero.  Our proposed method find the approximate solution by converting the integral equations into a linear system of algebraic equations which is easy to solve. The convergence of sequence of approximate solutions is proved and error bound is obtained theoretically. The validation of derived theoretical results and implementation of method is also shown with the aid of numerical illustrations.                            

APPROXIMATE SOLUTIONS OF NONLINEAR VOLTERRA INTEGRAL EQUATION SYSTEMS

2010 ◽  
Vol 24 (32) ◽  
pp. 6235-6258 ◽  
Author(s):  
SALIH YALÇINBAŞ ◽  
KÜBRA ERDEM
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Volterra Integral Equation ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Nonlinear Volterra Integral Equation ◽  
Solution Function ◽  
Taylor Coefficients ◽  
Linear Algebraic Equations ◽  
New System

The purpose of this study is to implement a new approximate method for solving system of nonlinear Volterra integral equations. The technique is based on, first, differentiating both sides of integral equations n times and then substituting the Taylor series the unknown functions in the resulting equation and later, transforming to a matrix equation. By merging these results, a new system which corresponds to a system of linear algebraic equations is obtained. The solution of this system yields the Taylor coefficients of the solution function. Some numerical results are also given to illustrate the efficiency of the method.

scholarly journals Approximate Solutions of Time Fractional Diffusion Wave Models

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 923 ◽  
Author(s):  
Abdul Ghafoor ◽  
Sirajul Haq ◽  
Manzoor Hussain ◽  
Poom Kumam ◽  
Muhammad Asif Jan
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Quadrature Rule ◽  
Approximate Solutions ◽  
Fractional Diffusion ◽  
Haar Wavelets ◽  
Test Problems ◽  
Algebraic Equations ◽  
Diffusion Wave ◽  
Wave Models

In this paper, a wavelet based collocation method is formulated for an approximate solution of (1 + 1)- and (1 + 2)-dimensional time fractional diffusion wave equations. The main objective of this study is to combine the finite difference method with Haar wavelets. One and two dimensional Haar wavelets are used for the discretization of a spatial operator while time fractional derivative is approximated using second order finite difference and quadrature rule. The scheme has an excellent feature that converts a time fractional partial differential equation to a system of algebraic equations which can be solved easily. The suggested technique is applied to solve some test problems. The obtained results have been compared with existing results in the literature. Also, the accuracy of the scheme has been checked by computing L 2 and L ∞ error norms. Computations validate that the proposed method produces good results, which are comparable with exact solutions and those presented before.

A stochastic operational matrix method for numerical solutions of multi-dimensional stochastic Itô–Volterra integral equations

2020 ◽  
Vol 28 (3) ◽  
pp. 209-216
Author(s):  
S. Singh ◽  
S. Saha Ray
Keyword(s):  
Integral Equations ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Volterra Integral Equations ◽  
Numerical Examples ◽  
Operational Matrix Method ◽  
Block Pulse Functions ◽  
Stochastic Operational Matrix ◽  
Block Pulse Function

AbstractIn this article, hybrid Legendre block-pulse functions are implemented in determining the approximate solutions for multi-dimensional stochastic Itô–Volterra integral equations. The block-pulse function and the proposed scheme are used for deriving a methodology to obtain the stochastic operational matrix. Error and convergence analysis of the scheme is discussed. A brief discussion including numerical examples has been provided to justify the efficiency of the mentioned method.

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