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1.
Based on a new approach, we show that finding solutions for a class of systems of linear (respectively, nonlinear) Fredholm integral equations of the third kind with multipoint singularities is equivalent to finding solutions of systems of linear (respectively, nonlinear) Fredholm integral equations of the second kind with additional conditions. We study the existence, nonexistence, uniqueness, and nonuniqueness of solutions for this class of systems of Fredholm integral equations of the third kind with multipoint singularities. 相似文献
2.
A. A. Kashirin S. I. Smagin 《Computational Mathematics and Mathematical Physics》2012,52(8):1173-1185
Three-dimensional Dirichlet problems for the Helmholtz equation are considered in generalized formulations. By applying single-layer potentials, they are reduced to Fredholm boundary integral equations of the first kind. The equations are discretized using a special averaging method for integral operators with weak singularities in the kernels. As a result, the integral equations are approximated by systems of linear algebraic equations with easy-to-compute coefficients, which are solved numerically by applying the generalized minimal residual method. A modification of the method is proposed that yields solutions in the spectra of interior Dirichlet problems and integral operators when the integral equations are not equivalent to the original differential problems and are not well-posed. Numerical results are presented for assessing the capabilities of the approach. 相似文献
3.
Claus Schneider 《Integral Equations and Operator Theory》1979,2(1):62-68
Continuity and differentiability properties of the solution to a class of Fredholm integral equations of the second kind with weakly singular kernel are derived. The equations studied in this paper arise from e.g. potential problems or problems of radiative equilibrium. Under reasonable assumptions it is proved that the solution possesses continuous derivatives in the interior of the interval of integration but may have mild singularities at the end-points. 相似文献
4.
《Chaos, solitons, and fractals》2007,31(1):138-146
In this paper we use parametric form of fuzzy number and convert a linear fuzzy Fredholm integral equation to two linear system of integral equation of the second kind in crisp case. We can use one of the numerical method such as Nystrom and find the approximation solution of the system and hence obtain an approximation for fuzzy solution of the linear fuzzy Fredholm integral equations of the second kind. The proposed method is illustrated by solving some numerical examples. 相似文献
5.
P. A. Krutitskii 《Differential Equations》2008,44(9):1203-1217
The Dirichlet problem for the Stokes equations outside open curves on the plane is studied. The existence and uniqueness of a solution is proved. An integral representation for the solution is obtained in the form of potentials whose densities are determined from a uniquely solvable system of Fredholm integral equations of the second kind. The singularities of derivatives of the velocities at the endpoints of the open curves are analyzed. A closed-form solution of the problem is constructed for the case in which the open curves are straight line segments. 相似文献
6.
The behaviour of a solution to a Fredholm integral equation of the second kind on a union of open intervals is examined. The kernel of the corresponding integral operator may have diagonal singularities, information about them is given through certain estimates. The weighted spaces of smooth functions with boundary singularities containing the solution of the integral equation are described. 相似文献
7.
A. V. Voytishek 《Journal of Applied and Industrial Mathematics》2018,12(2):382-394
Under study are the randomized algorithms for numerical solution of the Fredholm integral equations of the second kind (from the viewpoint of their application for the practically important problems of mathematical physics). The projection, grid and projection-grid methods are distinguished. Certain advantages of the projection and projection-grid methods are demonstrated (allowing using them for numerical solution of the equations with the integrable singularities in kernels and free terms). 相似文献
8.
The author proposes a numerical procedure in order to approximate the solution of a class of Fredholm integral equations of the third kind on unbounded domains. The given equation is transformed in a Fredholm integral equation of the second kind. Hence, according to the integration interval, the equation is regularized by means of a suitable one-to-one map or is transformed in a system of two Fredholm integral equations that are subsequently regularized. In both cases a Nyström method is applied, the convergence and the stability of which are proved in spaces of weighted continuous functions. Error estimates and numerical tests are also included. 相似文献
9.
Tomoaki Okayama Takayasu Matsuo 《Journal of Computational and Applied Mathematics》2010,234(4):1211-3303
In this paper we propose new numerical methods for linear Fredholm integral equations of the second kind with weakly singular kernels. The methods are developed by means of the Sinc approximation with smoothing transformations, which is an effective technique against the singularities of the equations. Numerical examples show that the methods achieve exponential convergence, and in this sense the methods improve conventional results where only polynomial convergence have been reported so far. 相似文献
10.
A. A. Kashirin S. I. Smagin M. Yu. Taltykina 《Computational Mathematics and Mathematical Physics》2016,56(4):612-625
Interior and exterior three-dimensional Dirichlet problems for the Helmholtz equation are solved numerically. They are formulated as equivalent boundary Fredholm integral equations of the first kind and are approximated by systems of linear algebraic equations, which are then solved numerically by applying an iteration method. The mosaic-skeleton method is used to speed up the solution procedure. 相似文献
11.
Lothar Reichel Andriy Shyshkov 《Journal of Computational and Applied Mathematics》2010,233(5):1314-1325
Multilevel methods are popular for the solution of well-posed problems, such as certain boundary value problems for partial differential equations and Fredholm integral equations of the second kind. However, little is known about the behavior of multilevel methods when applied to the solution of linear ill-posed problems, such as Fredholm integral equations of the first kind, with a right-hand side that is contaminated by error. This paper shows that cascadic multilevel methods with a conjugate gradient-type method as basic iterative scheme are regularization methods. The iterations are terminated by a stopping rule based on the discrepancy principle. 相似文献
12.
Mohamed A. Ramadan Heba S. Osheba Adel R. Hadhoud 《Numerical Methods for Partial Differential Equations》2023,39(1):268-280
This paper deals with the numerical solution of the integral equations of linear second kind Volterra–Fredholm. These integral equations are commonly used in engineering and mathematical physics to solve many of the problems. A hybrid of Bernstein and improved block-pulse functions method is introduced and used where the key point is to transform linear second-type Volterra–Fredholm integral equations into an algebraic equation structure that can be solved using classical methods. Numeric examples are given which demonstrate the related features of the process. 相似文献
13.
We study the Dirichlet problem for the static elasticity equations outside several curvilinear cracks on the plane. The existence
and uniqueness of a classical solution are proved. We obtain an integral representation of the solution in the form of potentials
whose densities are determined from a uniquely solvable system of Fredholm integral equations of the second kind. We analyze
the singularities of the derivatives of the solution at the endpoints of open curves and obtain a closed-form solution of
the problem for the case in which all cracks are located along segments of one and the same straight line. 相似文献
14.
The problem of determining the Stokes flow of a micropolar fluid exterior to several closed surfaces but contained by an exterior contour that encloses all the interior surfaces, is formulated as a system of linear Fredholm integral equations of the second kind. These integral equations are obtained when the velocity and microrotation vector fields are represented by a double-layer potential with unknown density, and certain singular solutions of the Stokes' micropolar equations. This double-layer potential is defined over the union of all the surfaces involved including the exterior contour. The singularities, corresponding to a concentrated force and concentrated couple located within each interior surface, give rise to force and torque whose magnitudes are linearly dependent on the unknown density of the double layer. It is shown that the system possesses a unique continuous solution when the boundaries are Lyapunov surfaces and the boundary data is continuous. 相似文献
15.
N.I. Ioakimidis 《Applied mathematics and computation》1983,12(1):49-60
The collocation method for the numerical solution of Fredholm integral equations of the second kind is applied, properly modified, to the numerical solution of Cauchy type singular integral equations of the first or the second kind but with constant coefficients. This direct method of numerical solution of Cauchy type singular integral equations is compared afterwards with the corresponding method resulting from applying the collocation method to the Fredholm integral equation of the second kind equivalent to the Cauchy type singular integral equation, as well as with another method, based also on the regularization procedure, for the numerical solution of the same class of equations. Finally, the convergence of the method is discussed. 相似文献
16.
The main purpose of this article is to describe a numerical scheme for solving two-dimensional linear Fredholm integral equations of the second kind on a non-rectangular domain. The method approximates the solution by the discrete collocation method based on radial basis functions (RBFs) constructed on a set of disordered data. The proposed method does not require any background mesh or cell structures, so it is meshless and consequently independent of the geometry of domain. This approach reduces the solution of the two-dimensional integral equation to the solution of a linear system of algebraic equations. The error analysis of the method is provided. The proposed scheme is also extended to linear mixed Volterra–Fredholm integral equations. Finally, some numerical examples are presented to illustrate the efficiency and accuracy of the new technique. 相似文献
17.
A. A. Kashirin M. Yu. Taltykina 《Computational Mathematics and Mathematical Physics》2017,57(9):1404-1415
Exterior three-dimensional Dirichlet problems for the Laplace and Helmholtz equations are considered. By applying methods of potential theory, they are reduced to equivalent Fredholm boundary integral equations of the first kind, for which discrete analogues, i.e., systems of linear algebraic equations (SLAEs) are constructed. The existence of mosaic-skeleton approximations for the matrices of the indicated systems is proved. These approximations make it possible to reduce the computational complexity of an iterative solution of the SLAEs. Numerical experiments estimating the capabilities of the proposed approach are described. 相似文献
18.
A. Chakrabarti 《Applied mathematics and computation》2009,211(2):459-466
This note is concerned with the problem of determining approximate solutions of Fredholm integral equations of the second kind. Approximating the solution of a given integral equation by means of a polynomial, an over-determined system of linear algebraic equations is obtained involving the unknown coefficients, which is finally solved by using the least-squares method. Several examples are examined in detail. 相似文献
19.
We suggest a numerical method for solving systems of linear nonautonomous ordinary differential equations with nonseparated multipoint and integral conditions. By using this method, which is based on the operation of convolution of integral conditions into local ones, one can reduce the solution of the original problem to the solution of a Cauchy problem for systems of ordinary differential equations and linear algebraic equations. We establish bounded linear growth of the error of the suggested numerical schemes. Numerical experiments were carried out for specially constructed test problems. 相似文献
20.
M. Heydari Z. Avazzadeh H. Navabpour G.B. Loghmani 《Applied Mathematical Modelling》2013,37(1-2):432-442
In this work, we generalize the numerical method discussed in [Z. Avazzadeh, M. Heydari, G.B. Loghmani, Numerical solution of Fredholm integral equations of the second kind by using integral mean value theorem, Appl. math. modelling, 35 (2011) 2374–2383] for solving linear and nonlinear Fredholm integral and integro-differential equations of the second kind. The presented method can be used for solving integral equations in high dimensions. In this work, we describe the integral mean value method (IMVM) as the technical algorithm for solving high dimensional integral equations. The main idea in this method is applying the integral mean value theorem. However the mean value theorem is valid for multiple integrals, we apply one dimensional integral mean value theorem directly to fulfill required linearly independent equations. We solve some examples to investigate the applicability and simplicity of the method. The numerical results confirm that the method is efficient and simple. 相似文献