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1.
On the numerical solution of singular two‐point boundary value problems: A domain decomposition homotopy perturbation approach
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Pradip Roul 《Mathematical Methods in the Applied Sciences》2017,40(18):7396-7409
This paper reports a modified homotopy perturbation algorithm, called the domain decomposition homotopy perturbation method (DDHPM), for solving two‐point singular boundary value problems arising in science and engineering. The essence of the approach is to split the domain of the problem into a number of nonoverlapping subdomains. In each subdomain, a method based on a combination of HPM and integral equation formalism is implemented. The boundary condition at the right endpoint of each inner subdomain is established before deriving an iterative scheme for the components of the solution series. The accuracy and efficiency of the DDHPM are demonstrated by 4 examples (2 nonlinear and 2 linear). In comparison with the traditional HPM, the proposed domain decomposition HPM is highly accurate. 相似文献
2.
We propose and analyze a two‐level method of discretizing the nonlinear Navier‐Stokes equations with slip boundary condition. The slip boundary condition is appropriate for problems that involve free boundaries, flows past chemically reacting walls, and other examples where the usual no‐slip condition u = 0 is not valid. The two‐level algorithm consists of solving a small nonlinear system of equations on the coarse mesh and then using that solution to solve a larger linear system on the fine mesh. The two‐level method exploits the quadratic nonlinearity in the Navier‐Stokes equations. Our error estimates show that it has optimal order accuracy, provided that the best approximation to the true solution in the velocity and pressure spaces is bounded above by the data. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 26–42, 2001 相似文献
3.
Dulat Dzhumabaev Elmira Bakirova Sandugash Mynbayeva 《Mathematical Methods in the Applied Sciences》2020,43(4):1788-1802
A nonlinear loaded differential equation with a parameter on a finite interval is studied. The interval is partitioned by the load points, at which the values of the solution to the equation are set as additional parameters. A nonlinear boundary value problem for the considered equation is reduced to a nonlinear multipoint boundary value problem for the system of nonlinear ordinary differential equations with parameters. For fixed parameters, we obtain the Cauchy problems for ordinary differential equations on the subintervals. Substituting the values of the solutions to these problems into the boundary condition and continuity conditions at the partition points, we compose a system of nonlinear algebraic equations in parameters. A method of solving the boundary value problem with a parameter is proposed. The method is based on finding the solution to the system of nonlinear algebraic equations composed. 相似文献
4.
陈庆祥 《数学物理学报(B辑英文版)》1995,(1)
(陈庆祥)MIXEDBOUNDARYVALUEPROBLEMFORSECOND-ORDERSYSTEMOFDIFFERENTIALEQUATIONSOFTHEELLIPTICTYPE¥ChenQingxiang(Dept.ofMath.,Zhongs... 相似文献
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6.
《Communications in Nonlinear Science & Numerical Simulation》2011,16(9):3504-3511
A boundary value method for solving a class of nonlinear singularly perturbed two point boundary value problems with a boundary layer at one end is proposed. Using singular perturbation analysis the method consists of solving two problems; namely, a reduced problem and a boundary layer correction problem. We use Pade’ approximation to obtain the solution of the latter problem and to satisfy the condition at infinity. Numerical examples will be given to illustrate the method. 相似文献
7.
V. M. Sveshnikov A. O. Savchenko A. V. Petukhov 《Numerical Analysis and Applications》2018,11(4):346-358
We propose a method for solving three-dimensional boundary value problems for Laplace’s equation in an unbounded domain. It is based on non-overlapping decomposition of the exterior domain into two subdomains so that the initial problem is reduced to two subproblems, namely, exterior and interior boundary value problems on a sphere. To solve the exterior boundary value problem, we propose a singularity isolation method. To match the solutions on the interface between the subdomains (the sphere), we introduce a special operator equation approximated by a system of linear algebraic equations. This system is solved by iterative methods in Krylov subspaces. The performance of the method is illustrated by solving model problems. 相似文献
8.
Ibrahim Tekin Yashar T. Mehraliyev Mansur I. Ismailov 《Mathematical Methods in the Applied Sciences》2019,42(10):3739-3753
In this paper, an initial boundary value problem for nonlinear Klein‐Gordon equation is considered. Giving an additional condition, a time‐dependent coefficient multiplying nonlinear term is determined, and existence and uniqueness theorem for small times is proved. The finite difference method is proposed for solving the inverse problem. 相似文献
9.
A.S. Fokas 《Selecta Mathematica, New Series》1998,4(1):31-68
A new transform method for solving initial-boundary value problems for linear and integrable nonlinear PDEs in two independent
variables has been recently introduced in [1]. For linear PDEs this method involves: (a) formulating the given PDE as the
compatibility condition of two linear equations which, by analogy with the nonlinear theory, we call a Lax pair; (b) formulating
a classical mathematical problem, the so-called Riemann-Hilbert problem, by performing a simultaneous spectral analysis of both equations defining the Lax pair; (c) deriving certain global relations satisfied by the boundary
values of the solution of the given PDE. Here this method is used to solve certain problems for the heat equation, the linearized
Korteweg-deVries equation and the Laplace equation. Some of these problems illustrate that the new method can be effectively
used for problems with complicated boundary conditions such as changing type as well as nonseparable boundary conditions. It is shown that for simple boundary conditions the global relations (c) can be analyzed using only
algebraic manipulations, while for complicated boundary conditions, one needs to solve an additional Riemann-Hilbert problem.
The relationship of this problem with the classical Wiener-Hopf technique is pointed out. The extension of the above results
to integrable nonlinear equations is also discussed. In particular, the Korteweg-deVries equation in the quarter plane is
linearized. 相似文献
10.
Mourad Bellassoued Mourad Choulli 《Journal of Mathematical Analysis and Applications》2008,343(1):328-336
We establish a stability estimate for an inverse boundary coefficient problem in thermal imaging. The inverse problem under consideration consists in the determination of a boundary coefficient appearing in a boundary value problem for the heat equation with Robin boundary condition (we note here that the initial condition is assumed to be a priori unknown). Our stability estimate is of logarithmic type and it is essentially based on a logarithmic estimate for a Cauchy problem for the Laplace equation. 相似文献
11.
The first three-dimensional boundary value problem is considered for the basic equations of statics of the elastic mixture theory in the finite and infinite domains bounded by the closed surfaces. It is proved that this problem splits into two problems whose investigation is reduced to the first boundary value problem for an elliptic equation which structurally coincides with an equation of statics of an isotropic elastic body. Using the potential method and the theory of Fredholm integral equations of second kind, the existence and uniqueness of the solution of the first boundary value problem is proved for the split equation. 相似文献
12.
We establish the wellposedness of the time‐independent Navier–Stokes equations with threshold slip boundary conditions in bounded domains. The boundary condition is a generalization of Navier's slip condition and a restricted Coulomb‐type friction condition: for wall slip to occur the magnitude of the tangential traction must exceed a prescribed threshold, independent of the normal stress, and where slip occurs the tangential traction is equal to a prescribed, possibly nonlinear, function of the slip velocity. In addition, a Dirichlet condition is imposed on a component of the boundary if the domain is rotationally symmetric. We formulate the boundary‐value problem as a variational inequality and then use the Galerkin method and fixed point arguments to prove the existence of a weak solution under suitable regularity assumptions and restrictions on the size of the data. We also prove the uniqueness of the solution and its continuous dependence on the data. Copyright © 2006 John Wiley & Sons, Ltd. 相似文献
13.
We consider a symmetric Galerkin boundary element method for the Stokes problem with general boundary conditions including slip conditions. The boundary value problem is reformulated as Steklov–Poincaré boundary integral equation which is then solved by a standard approximation scheme. An essential tool in our approach is the invertibility of the single layer potential which requires the definition of appropriate factor spaces due to the topology of the domain. Here we describe a modified boundary element approach to solve Dirichlet boundary value problems in multiple connected domains. A suitable extension of the standard single layer potential leads to an operator which is elliptic on the original function space. Copyright © 2003 John Wiley & Sons, Ltd. 相似文献
14.
We continue our study of the construction of numerical methods for solving two-point boundary value problems using Green functions, building on the successful use of split-Gauss-type quadrature schemes. Here we adapt the method for eigenvalue problems, in particular the Orr–Sommerfeld equation of hydrodynamic stability theory. Use of the Green function for the viscous part of the problem reduces the fourth-order ordinary differential equation to an integro-differential equation which we then discretize using the split-Gaussian quadrature and product integration approach of our earlier work along with pseudospectral differentiation matrices for the remaining differential operators. As the latter are only second-order the resulting discrete equations are much more stable than those obtained from the original differential equation. This permits us to obtain results for the standard test problem (plane Poiseuille flow at unit streamwise wavenumber and Reynolds number 10 000) that we believe are the most accurate to date. 相似文献
15.
Hari M. Srivastava Anvar Hasanov Tuhtasin Gulamjanovich Ergashev 《Mathematical Methods in the Applied Sciences》2020,43(10):6181-6199
Potentials play an important role in solving boundary value problems for elliptic equations. In the middle of the last century, a potential theory was constructed for a two-dimensional elliptic equation with one singular coefficient. In the study of potentials, the properties of the fundamental solutions of the given equation are essentially and fruitfully used. At the present time, fundamental solutions of a multidimensional elliptic equation with one degeneration line are already known. In this paper, we investigate the double- and simple-layer potentials for this kind of elliptic equations. Results from potential theory allow us to represent the solution of the boundary value problems in the form of an integral equation. By using some properties of the Gaussian hypergeometric function, we first prove limiting theorems and derive integral equations concerning the densities of the double- and simple-layer potentials. The obtained results are then applied in order to find an explicit solution of the Holmgren problem for the multidimensional singular elliptic equation in the half of the ball. 相似文献
16.
Shengfeng Zhu Qingbiao Wu Xiaoliang Cheng 《Applied mathematics and computation》2009,215(7):2472-2485
We present two iterative methods for solving the Falkner-Skan equation based on the quasilinearization method. We formulate the original problem as a new free boundary value problem. The truncated boundary depending on a small parameter is an unknown free boundary and has to be determined as part of solution. Using a change of variables, the free boundary value problem is transformed to a problem defined on [0, 1]. We apply the quasilinearization method to solve the resulting nonlinear problem. Then we propose two different iterative algorithms by means of a cubic spline solver. Numerical results for various instances are compared with those reported previously in the literature. The comparisons show the accuracy, robustness and efficiency of the presented methodology. 相似文献
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18.
In this paper, we consider the Dirichlet and impedance boundary value problems for the Helmholtz equation in a non‐locally perturbed half‐plane. These boundary value problems arise in a study of time‐harmonic acoustic scattering of an incident field by a sound‐soft, infinite rough surface where the total field vanishes (the Dirichlet problem) or by an infinite, impedance rough surface where the total field satisfies a homogeneous impedance condition (the impedance problem). We propose a new boundary integral equation formulation for the Dirichlet problem, utilizing a combined double‐ and single‐layer potential and a Dirichlet half‐plane Green's function. For the impedance problem we propose two boundary integral equation formulations, both using a half‐plane impedance Green's function, the first derived from Green's representation theorem, and the second arising from seeking the solution as a single‐layer potential. We show that all the integral equations proposed are uniquely solvable in the space of bounded and continuous functions for all wavenumbers. As an important corollary we prove that, for a variety of incident fields including an incident plane wave, the impedance boundary value problem for the scattered field has a unique solution under certain constraints on the boundary impedance. Copyright © 2003 John Wiley & Sons, Ltd. 相似文献
19.
This paper developed an analytical solution for the problem of exit point evolution on the seepage face in the unconfined aquifer with sloping interface. A theoretical model for the groundwater drawdown problem in a half‐infinite aquifer with a sloping boundary is built in accordance with the linearized one‐dimensional Boussinesq equation and the Neumann boundary condition at the seepage point. The homotopy analysis method is then adopted for solving this dynamic boundary problem. By constructing two continuous deformations, the original problem could be converted into a group of subproblems with the same physical essence and similar mathematical solutions. To compare this analytical solution, a numerical model based on the finite volume method is developed, which employs adaptive grids to settle the dynamic boundary condition. The comparisons show that the analytical solution agrees with the numerical model well. The results are useful for the quantification of various hydrological problems. The methodology applied in this study is referential for other dynamic boundary problems as well. 相似文献
20.
We solve a Dirichlet boundary value problem for the Klein–Gordon equation posed in a time‐dependent domain. Our approach is based on a general transform method for solving boundary value problems for linear and integrable nonlinear PDE in two variables. Our results consist of the inversion formula for a generalized Fourier transform, and of the application of this generalized transform to the solution of the boundary value problem. 相似文献