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1.
Numerical solution of the fractional Bagley‐Torvik equation by using hybrid functions approximation
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In this paper, a new numerical method for solving the fractional Bagley‐Torvik equation is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block‐pulse functions and Bernoulli polynomials are presented. The Riemann‐Liouville fractional integral operator for hybrid functions is introduced. This operator is then utilized to reduce the solution of the initial and boundary value problems for the fractional Bagley‐Torvik differential equation to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
2.
Vladimir Varlamov 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2005,196(3):957-985
The Ostrovsky equation governs the propagation of long nonlinear surface waves in the presence of rotation. It is related
to the Korteweg-de Vries (KdV) and the Kadomtsev-Petviashvili models. KdV can be obtained from the equation in question when
the rotation parameter γ equals zero. A fundamental solution of the Cauchy problem for the linear Ostrovsky equation is presented
in the form of an oscillatory Fourier integral. Another integral representation involving Airy and Bessel functions is derived
for it. It is shown that its asymptotic expansion as γ → 0 contains the KdV fundamental solution as the zero term. The Airy
transform is used to establish some of its properties. Higher-order asymptotics for γ → 0 on a bounded time interval are obtained
for both the fundamental solution and the solution of the linear Cauchy problem for the Ostrovsky equation. 相似文献
3.
4.
Vladimir Varlamov 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2005,56(6):957-985
The Ostrovsky equation governs the propagation of long nonlinear surface waves in the presence of rotation. It is related
to the Korteweg-de Vries (KdV) and the Kadomtsev-Petviashvili models. KdV can be obtained from the equation in question when
the rotation parameter γ equals zero. A fundamental solution of the Cauchy problem for the linear Ostrovsky equation is presented
in the form of an oscillatory Fourier integral. Another integral representation involving Airy and Bessel functions is derived
for it. It is shown that its asymptotic expansion as γ → 0 contains the KdV fundamental solution as the zero term. The Airy
transform is used to establish some of its properties. Higher-order asymptotics for γ → 0 on a bounded time interval are obtained
for both the fundamental solution and the solution of the linear Cauchy problem for the Ostrovsky equation.
Received: November 23, 2004; revised: March 13, 2005
Research is supported by US Department of Defense, under grant No. DAAD19-03-1-0204 相似文献
5.
Andreas Rathsfeld 《Mathematical Methods in the Applied Sciences》1999,22(3):217-241
In the present paper we consider the radiosity equation over the boundary of a polyhedral domain. Similarly to corresponding results on the double‐layer potential equation, the solution of the second kind integral equation with non‐compact integral operator is piecewise continuous. The partial derivatives, however, are not bounded. In the present paper we derive the first term in the asymptotic expansion of the solution in the vicinity of an edge. Note that, knowing this term, optimal mesh gradings can be designed for the numerical solution of this equation. Copyright © 1999 John Wiley & Sons, Ltd. 相似文献
6.
Ricardo Cortez 《纯数学与应用数学通讯》1997,50(12):1235-1260
A proof of high-order convergence of three deterministic particle methods for the convection-diffusion equation in two dimensions is presented. The methods are based on discretizations of an integro-differential equation in which an integral operator approximates the diffusion operator. The methods differ in the discretization of this operator. The conditions for convergence imposed on the kernel that defines the integral operator include moment conditions and a condition on the kernel's Fourier transform. Explicit formulae for kernels that satisfy these conditions to arbitrary order are presented. © 1997 John Wiley & Sons, Inc. 相似文献
7.
M. Y. Neklyudov 《Journal of Mathematical Sciences》2008,150(6):2531-2539
In this paper, we prove the equivalence of the Navier-Stokes equation and the infinite-dimensional Burgers-type equation with
the Laplace-Lévy operator. An explicit formula for the solution of a certain system of linear equations arising in studying
the circulation of the solution of the Navier-Stokes equation is presented.
__________
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 5, pp. 109–120, 2006. 相似文献
8.
G. R. Thomson 《Mathematical Methods in the Applied Sciences》2015,38(17):4088-4096
Integral equations of the first kind for exterior problems arising in the study of the three‐dimensional Helmholtz equation are considered. These equations are derived by seeking solutions in the form of layer potentials with modified fundamental solutions. For each first kind equation, existence and uniqueness of solution are proved with the aid of composition relations involving associated modified boundary integral operators. For the Dirichlet problem, an optimal choice of the modification coefficients is considered in order to minimize the condition number of the resulting integral operator. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
9.
A concept of a fundamental solution is introduced for linear operator equations given in some functional spaces. In the case where this fundamental solution does not exist, the representation of the solution is found by the concept of a generalized fundamental solution, which is introduced for operators with nontrivial and generally infinite-dimensional kernels. The fundamental and generalized fundamental solutions are also investigated for a class of Fredholm-type operator equations. Some applications are given for one-dimensional generally nonlocal hyperbolic problems with trivial, finite- and infinite-dimensional kernels. The fundamental and generalized fundamental solutions of such problems are constructed as particular solutions of a system of integral equations or an integral equation. These fundamental solutions become meaningful in a general case when the coefficients are generally nonsmooth functions satisfying only some conditions such as p-integrablity and boundedness. 相似文献
10.
Using a standard application of Green's theorem, the exterior Dirichlet problem for the Laplace equation in three dimensions is reduced to a pair of integral equations. One integral equation is of the second kind and the other is of the first kind. It is known that the integral equation of the second kind is not uniquely solvable, however, it has been demonstrated that the pair of integral equations has a unique solution. The present approach is based on the observation that the known function appearing in the integral equation of the second kind lies in a certain Banach space E which is a proper subspace of the Banach space of continuous complex-valued functions equipped with the maximum norm. Furthermore, it can be shown that the related integral operator when restricted to E has spectral radius less than unity. Consequently, a particular solution to the integral equation of the second kind can be obtained by the method of successive approximations and the unique solution to the problem is then obtained by using the integral equation of the first kind. Comparisons are made between the present algorithm and other known constructive methods. Finally, an example is supplied to illustrate the method of this paper. 相似文献
11.
This paper presents a mathematical modeling and a numerical methodological approach based on integral equation formulation and radial basis functions (RBF) for the dynamic behaviour damped Timoshenko beams under follower forces. Using the fundamental solution of the main operator and the RBF, the governing non-self-adjoint partial differential equation is transformed into an integral equation. Based on the harmonic assumption and internal concatenation points an eigenvalue problem is obtained and numerically solved. The flutter analysis, complex modes and the frequency load responses are investigated for beams under various subtangential follower loads, aspect ratios, internal and viscous damping. 相似文献
12.
泊松类型方程边界元解法 总被引:2,自引:0,他引:2
本文采用高阶拉普拉斯算子基本解将泊松类型方程的区域积分全部变换成边界积分,使计算问题的维数减少一维.通过斯托克斯方程的算例,表明本文所用的方法是有效的方法。 相似文献
13.
We consider a linear integral equation with a hypersingular integral treated in the sense of the Hadamard finite value. This
equation arises in the solution of the Neumann boundary value problem for the Laplace equation with a representation of a
solution in the form of a double-layer potential. We consider the case in which the interior or exterior boundary value problem
is solved in a domain; whose boundary is a smooth closed surface, and an integral equation is written out on that surface.
For the integral operator in that equation, we suggest quadrature formulas like the method of vortical frames with a regularization,
which provides its approximation on the entire surface for the use of a nonstructured partition. We construct a numerical
scheme for the integral equation on the basis of suggested quadrature formulas, prove an estimate for the norm of the inverse
matrix of the related system of linear equations and the uniform convergence of numerical solutions to the exact solution
of the hypersingular integral equation on the grid. 相似文献
14.
An inverse problem of the reconstruction of the right-hand side of the Euler-Darboux equation is studied. This problem is equivalent to the Volterra integral equation of the third kind with the operator of multiplication by a smooth nonincreasing function. Numerical solution of this problem is constructed using an integral representation of the solution of the inverse problem, the regularization method, and the method of quadratures. The convergence and stability of the numerical method is proved. 相似文献
15.
E. H. Khalilov 《Computational Mathematics and Mathematical Physics》2016,56(7):1310-1318
The surface integral equation for a spatial mixed boundary value problem for the Helmholtz equation is considered. At a set of chosen points, the equation is replaced with a system of algebraic equations, and the existence and uniqueness of the solution of this system is established. The convergence of the solutions of this system to the exact solution of the integral equation is proven, and the convergence rate of the method is determined. 相似文献
16.
《Journal of Mathematical Analysis and Applications》1987,126(2):547-555
A boundary integral equation for the exterior Robin problem for Helmholtz's equation is analyzed in this paper. This integral operator is not compact. A proof based on a suitable regularization of this integral operator and the Fredholm alternative for the regularized compact operator was given by other authors. In this paper, we will give a direct existence and uniqueness proof for the boundary non-compact integral equation in the space settings C1,λ(S) and C0,λ(S), where S is a closed bounded smooth surface. 相似文献
17.
Numerical analysis of boundary integral solution of the Helmholtz equation in domains with non-smooth boundaries 总被引:1,自引:0,他引:1
In the paper we carry out a complete analysis of several efficientnumerical methods for the solution of boundary integral equationsdefined on a non-smooth boundary. In particular the solutionof the Helmholtz equation in the exterior of a closed wedgeis studied. The analytical behaviour of the solution of theresulting boundary integral equation (with a non-compact operator)near the wedge is investigated. Numerical analysis of the collocationand iterated collocation method for the problem is presented.Graded meshes are used to reflect the singularbehaviour of the analytical solution, as well as the degreeof the polynomial approximant, in order to yield results withoptimal convergence rates. Finally the convergenceanalysis of some modified two-grid iterative methods for thefast solution of the resulting linear systems is given and numericalresults are presented which agree with the theoretical predictions. 相似文献
18.
Karen Yagdjian 《Journal of Differential Equations》2004,206(1):227-252
Barros-Neto and Gelfand (Duke Math. J. 98 (3) (1999) 465; Duke Math. J. 117 (2) (2003) 561) constructed for the Tricomi operator on the plane the fundamental solutions with the supports in the regions related to the geometry of the characteristics of the Tricomi operator. We give for the Tricomi-type operator a fundamental solution relative to an arbitrary point of Rn+1 with the support in the region t?0, where the operator is hyperbolic. Our key observation is that the fundamental solution for the Tricomi-type operator can be written like an integral of the distributions generated by the fundamental solution of the Cauchy problem for the wave equation. The application of that fundamental solution to the Lp-Lq estimate for the forced Tricomi-type equation is given as well. 相似文献
19.
The solution of a convolution-type equation for two unknown compactly supported functions for the case in which the kernel
of the integral operator has Fourier transform in the form of the ratio of two polynomials of exponentials and the right-hand
side satisfies additional conditions like the evenness-oddness, absence of zeros, etc. is reduced to the solution of a system
of difference equations. We suggest a solution method for that system. 相似文献
20.
《Mathematical Methods in the Applied Sciences》2018,41(4):1316-1330
We propose a new integral equation formulation to characterize and compute transmission eigenvalues in electromagnetic scattering. As opposed to the approach that was recently developed by Cakoni, Haddar and Meng (2015) which relies on a two‐by‐two system of boundary integral equations, our analysis is based on only one integral equation in terms of the electric‐to‐magnetic boundary trace operator that results in a simplification of the theory and in a considerable reduction of computational costs. We establish Fredholm properties of the integral operators and their analytic dependence on the wave number. Further, we use the numerical algorithm for analytic nonlinear eigenvalue problems that was recently proposed by Beyn (2012) for the numerical computation of the transmission eigenvalues via this new integral equation. 相似文献