scholarly journals Use of the multigrid methods for heat radiation problem

2003 ◽  
Vol 2003 (6) ◽  
pp. 305-317 ◽  
Author(s):  
Naji A. Qatanani
Keyword(s):  
Integral Equation ◽  
High Efficiency ◽  
Linear Equations ◽  
Numerical Realization ◽  
Heat Radiation ◽  
System Of Linear Equations ◽  
Radiation Problem ◽  
Radiation Exchange ◽  
Efficient Detection ◽  
Iteration Methods

We consider the integral equation arising as a result of heat radiation exchange in both convex and nonconvex enclosures of diffuse grey surfaces. For nonconvex geometries, the visibility function must be taken into consideration. Therefore, a geometrical algorithm has been developed to provide an efficient detection of the shadow zones. For the numerical realization of the Fredholm integral equation, a boundary element method based on Galerkin-Bubnov discretization scheme is implemented. Consequently, multigrid iteration methods, which are closely related to two-grid methods, are used to solve the system of linear equations. To demonstrate the high efficiency of these iterations, we construct some numerical experiments for different enclosure geometries.

scholarly journals Hybrid Bernstein Block-Pulse Functions Method for Second Kind Integral Equations with Convergence Analysis

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Mohsen Alipour ◽  
Dumitru Baleanu ◽  
Fereshteh Babaei
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Convergence Analysis ◽  
Bernstein Polynomials ◽  
Linear Equations ◽  
The Other ◽  
New Combination ◽  
System Of Linear Equations ◽  
Numerical Examples ◽  
Block Pulse Functions

We introduce a new combination of Bernstein polynomials (BPs) and Block-Pulse functions (BPFs) on the interval [0, 1]. These functions are suitable for finding an approximate solution of the second kind integral equation. We call this method Hybrid Bernstein Block-Pulse Functions Method (HBBPFM). This method is very simple such that an integral equation is reduced to a system of linear equations. On the other hand, convergence analysis for this method is discussed. The method is computationally very simple and attractive so that numerical examples illustrate the efficiency and accuracy of this method.

Introducing new global electromagnetic modeling solver

2020 ◽  
Author(s):  
Mikhail Kruglyakov ◽  
Alexey Kuvshinov
Keyword(s):  
Integral Equation ◽  
Galerkin Method ◽  
Numerical Solution ◽  
Linear Equations ◽  
Numerical Algorithms ◽  
Iterative Solution ◽  
Parallel Computations ◽  
Electromagnetic Modeling ◽  
System Of Linear Equations

<p> In this contribution, we present novel global 3-D electromagnetic forward solver based on a numerical solution of integral equation (IE) with contracting kernel. Compared to widely used x3dg code which is also based on IE approach, new solver exploits alternative (more efficient and accurate) numerical algorithms to calculate Green’s tensors, as well as an alternative (Galerkin) method to construct the system of linear equations (SLE). The latter provides guaranteed convergence of the iterative solution of SLE. The solver outperforms x3dg in terms of accuracy, and, in contrast to (sequential) x3dg, it allows for efficient parallel computations, meaning that the code has practically linear scalability up to the hundreds of processors.</p>

THE ELECTROSTATIC FIELD OF TWO EQUAL CIRCULAR CO-AXIAL CONDUCTING DISKS

1949 ◽  
Vol 2 (4) ◽  
pp. 428-451 ◽  
Author(s):  
E. R. LOVE
Keyword(s):  
Integral Equation ◽  
Explicit Solution ◽  
Existence And Uniqueness ◽  
Infinite System ◽  
Linear Equations ◽  
Neumann Series ◽  
Standard Theory ◽  
System Of Linear Equations ◽  
Electrostatic Problem ◽  
Spheroidal Harmonics

Abstract In the earliest discussion of this problem Nicholson (1) expressed the potential as a series of spheroidal harmonics with coefficients satisfying an infinite system of linear equations, and gave a formula for an explicit solution; but this formula appears to be meaningless and its derivation to contain serious errors. In the present paper, starting tentatively from Nicholson's infinite system of linear equations, a much simpler, though still implicit, specification of the potential is developed; this involves a Fredholm integral equation the existence and uniqueness of whose solution are deducible from standard theory. The specification so obtained for the potential is shown rigorously to satisfy the differential equation and boundary conditions of the electrostatic problem. The Neumann series of the integral equation is shown to converge to its solution, so that the potential, and other aspects of the field, can be explicitly formulated and thus computed. The errors in Nicholson's process of solving his system of equations are exhibited in detail, and it is concluded that attempts to carry through that process without error cannot lead to an explicit solution.

scholarly journals Publicly verifiable and efficiency/security-adjustable outsourcing scheme for solving large-scale modular system of linear equations

2019 ◽  
Vol 8 (1) ◽  
Author(s):  
Panpan Meng ◽  
Chengliang Tian ◽  
Xiangguo Cheng
Keyword(s):  
Coding Theory ◽  
Large Scale ◽  
High Efficiency ◽  
Linear Equations ◽  
Coefficient Matrix ◽  
Secure Computation ◽  
Modular System ◽  
System Of Linear Equations ◽  
Public Verifiability ◽  
Cloud Server

AbstractSolving large-scale modular system of linear equations ($\mathcal {LMSLE}$ℒℳSℒE) is pervasive in modern computer and communication community, especially in the fields of coding theory and cryptography. However, it is computationally overloaded for lightweight devices arisen in quantity with the dawn of the things of internet (IoT) era. As an important form of cloud computing services, secure computation outsourcing has become a popular topic. In this paper, we design an efficient outsourcing scheme that enables the resource-constrained client to find a solution of the $\mathcal {LMSLE}$ℒℳSℒE with the assistance of a public cloud server. By utilizing affine transformation based on sparse unimodular matrices, our scheme has three merits compared with previous work: 1) Our scheme is efficiency/security-adjustable. Our encryption method is dynamic, and it can balance the security and efficiency to match different application scenarios by skillfully control the number of unimodular matrices. 2) Our scheme is versatile. It is suit for generic m-by-n coefficient matrix A, no matter it is square or not. 3) Our scheme satisfies public verifiability and achieves the optimal verification probability. It enables any verifier which is not necessarily the client to verify the correctness of the results returned from the cloud server with probability 1. Finally, theoretical analysis and comprehensive experimental results confirm our scheme’s security and high efficiency.

scholarly journals Integral equation with the generalized neumann kernel for computing green’s function on simply connected regions

2014 ◽  
Vol 9 (3) ◽  
Author(s):  
Ali H. M. Murid ◽  
Mohmed M. A. Alagele ◽  
Mohamed M. S. Nasser
Keyword(s):  
Integral Equation ◽  
Green’S Function ◽  
Green's Function ◽  
Gaussian Elimination ◽  
Linear Equations ◽  
Boundary Integral ◽  
System Of Linear Equations ◽  
Generalized Neumann Kernel ◽  
Simply Connected ◽  
Neumann Kernel

This research is about computing the Green’s functions on simply connected regions by using the method of boundary integral equation. The method depends on solving a Dirichlet problem using a uniquely solvable Fredholm integral equation on the boundary of the region. The kernel of this integral equation is the generalized Neumann kernel. The numerical method for solving this integral equation is the Nystrӧm method with trapezoidal rule which leads to a system of linear equations. The linear system is then solved by the Gaussian elimination method. Mathematica plot of Green’s function for atest region is also presented.

scholarly journals A Numerical Solution of Fredholm Integral Equations of the Second Kind Based on Tight Framelets Generated by the Oblique Extension Principle

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 854 ◽  
Author(s):  
Mutaz Mohammad
Keyword(s):  
Integral Equation ◽  
Numerical Solution ◽  
Integral Equations ◽  
Linear Equations ◽  
Computational Method ◽  
Fredholm Integral Equations ◽  
Extension Principle ◽  
System Of Linear Equations ◽  
B Spline ◽  
Extension Principles

In this paper, we present a new computational method for solving linear Fredholm integral equations of the second kind, which is based on the use of B-spline quasi-affine tight framelet systems generated by the unitary and oblique extension principles. We convert the integral equation to a system of linear equations. We provide an example of the construction of quasi-affine tight framelet systems. We also give some numerical evidence to illustrate our method. The numerical results confirm that the method is efficient, very effective and accurate.

scholarly journals UNAMBIGUOUS SOLVABILITY OF A PARTICULAR CASE OF SYSTEMS OF INTEGRODIFFERENTIAL EQUATIONS WITH A PULSED KAEV DISTANCE CONTAINING A PARAMETER

2020 ◽  
Vol 72 (4) ◽  
pp. 78-84
Author(s):  
Kh.I. Usmanov ◽  
◽  
A.S. Zhappar ◽  
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Integral Equation ◽  
Cauchy Problem ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Linear Equations ◽  
Sufficient Conditions ◽  
System Of Linear Equations ◽  
New Variables

We consider a special case of systems of integro-differential equations with a momentum boundary condition containing a parameter when the derivative of the desired function is contained in the right side of the equation. By integrating in parts, an integro-differential equation with a pulsed boundary condition is reduced to a loaded integrodifferential equation with a pulsed boundary condition. it is given in the system of integral-differential equations with impulse boundary conditions parametrically loaded. Then, by entering new parameters, as well as passing to new variables based on these parameters, the problem is reduced to an equivalent problem. Switching to new variables makes it possible to get the initial conditions for the equation. Based on this, the solution of the problem is reduced to solving a special Cauchy problem and a system of linear equations. Using the fundamental matrix of the main part of the differential equation, an integral equation of the Volterra type is obtained. The method of sequential approximation determines the unique solution of the integral equation. Based on this, we find a solution to the special Cauchy problem and put it in the boundary conditions. On the basis of the obtained system of linear equations, necessary and sufficient conditions for an unambiguous solution of the initial problem are established.

Elastic Contact of a Strip Pressed Between Two Cylinders

1968 ◽  
Vol 35 (2) ◽  
pp. 279-284 ◽  
Author(s):  
C. F. Wang
Keyword(s):  
Integral Equation ◽  
Contact Problem ◽  
Linear Equations ◽  
Approximate Solutions ◽  
Thin Layers ◽  
Circular Cylinders ◽  
Elastic Contact ◽  
System Of Linear Equations ◽  
Fourier Cosine ◽  
Cosine Transformation

Methods of solving an integral equation that represents the elastic contact of a strip pressed between two identical cylinders are discussed. Fourier cosine transformation is used to derive the integral equation of the contact problem, and approximation kernels are used to obtain the solutions for the cases of thick and thin layers. The solutions for both cases are given as a truncated series of the weighted Tchebyshev polynomials of the first kind, whose coefficients are determined from a system of linear equations. The problem of two circular cylinders pressing on a strip is given as an example. The nature of two approximate solutions are also briefly discussed.

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