Auxiliary equation method for the mKdV equation with variable coefficients

By using solutions of an ordinary differential equation, an auxiliary equation method is described to seek exact solutions of nonlinear evolution equations with variable coefficients. Being concise and straightforward, this method is applied to the mKdV equation with variable coefficients. As a result, new explicit solutions including solitary wave solutions and trigonometric function solutions are obtained with the aid of symbolic computation.     

Convergence of the Galyorkin method of approximate solving parabolic equation with weight integral condition

In a Hilbert space an abstract linear parabolic equation with a nonlocal weight integral condition is resolved approximatelymaking use of theGalyorkinmethod. Assumptions on projection subspaces are oriented on a usage of finite element method. We consider the case of projection subspaces built by the uniform partition of domain as well as the case of arbitrary projection subspaces. We obtain the errors estimations for approximate solutions and establish estimates of the convergence rate, exact by the order.     

Approximate solutions of linear Volterra integral equation systems with variable coefficients

In this paper, a new approximate method has been presented to solve the linear Volterra integral equation systems (VIEs). This method transforms the integral system into the matrix equation with the help of Taylor series. 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. Also, this method gives the analytic solution when the exact solutions are polynomials. So as to show this capability and robustness, some systems of VIEs are solved by the presented method in order to obtain their approximate solutions.     

Reproducing Kernel method for the solution of nonlinear hyperbolic telegraph equation with an integral condition

In this article, an iterative method is proposed for solving nonlinear hyperbolic telegraph equation with an integral condition. Its exact solution is represented in the form of series in the reproducing kernel space. In the mean time, the n‐term approximation u_n(x, t) of the exact solution u(x, t) is obtained and is proved to converge to the exact solution. Moreover, the partial derivatives of u_n(x, t) are also convergent to the partial derivatives of u(x, t). Some numerical examples have been studied to demonstrate the accuracy of the present method. Results obtained by the method have been compared with the exact solution of each example and are found to be in good agreement with each other. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 867–886, 2011     

Cauchy problem for an essentially infinite-dimensional parabolic equation with variable coefficients

The Cauchy problem for the equation with positive essentially infinite-dimensional functionalsj(x) is studied in a properly chosen Banach space of functions on an infinite-dimensional separable real Hilbert space. Kiev Polytechnic Institute, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 6, pp. 663–670, June, 1994.     

Asymptotic behavior and uniqueness for an ultrahyperbolic equation with variable coefficients

This paper describes the asymptotic behavior of solutions of a class of semilinear ultrahyperbolic equations with variable coefficients. One consequence of the general analysis is a uniqueness theorem for a mixed boundary-value problem. Another demonstrates unique continuation at infinity. These results extend previous work by M. H. Protter, [Asymptotic decay for ultrahyperbolic operators, in “Contributions to Analysis” (Lars Ahlfors et al., Eds.), Academic Press, New York, 1974], and A. C. Murray and M. M. Protter, [Indiana U. Math. J.24 (1974), 115–130], on a more restricted class of equations.     

Wavelet method for a class of fractional convection-diffusion equation with variable coefficients

A wavelet method to the solution for a class of space–time fractional convection-diffusion equation with variable coefficients is proposed, by which combining Haar wavelet and operational matrix together and dispersing the coefficients efficaciously. The original problem is translated into Sylvester equation and computation became convenient. The numerical example shows that the method is effective.     

Exact controllability for the wave equation with variable coefficients

We consider in this paper the evolution systemy″−Ay=0, whereA =∂ _i(a_ij∂_j) anda _ij ∈C ¹ (ℝ⁺;W ^1,∞ (Ω)) ∩W ^1,∞ (Ω × ℝ⁺), with initial data given by (y ₀,y ₁) ∈L ²(Ω) ×H ⁻¹ (Ω) and the nonhomogeneous conditiony=v on Γ ×]0,T[. Exact controllability means that there exist a timeT>0 and a controlv such thaty(T, v)=y′(T, v)=0. The main result of this paper is to prove that the above system is exactly controllable whenT is “sufficiently large”. Moreover, we obtain sharper estimates onT.     

On the estimates for the number of solutions of an equation with integral coefficients

Translated from Matematicheskie Zametki, Vol. 54, No. 5, pp. 57–64, November, 1993.     

Beam evolution equation with variable coefficients

We investigate a initial‐boundary value problem for the nonlinear beam equation with variable coefficients on the action of a linear internal damping. We show the existence of a unique global weak solution and that the energy associated with this solution has a rate decay estimate. Besides, we prove the existence and uniqueness of non‐local strong solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

Alternating direction method for bi-quadratic programming

Bi-quadratic programming (Bi-QP for short) was studied systematically in Ling et al. (SIAM J. Optim. 20:1286–1320, 2009) due to its various applications in engineering as well as optimization. Several approximation methods were given in the same paper since it is NP-hard. In this paper, we introduce a quadratic SDP relaxation of Bi-QP and discuss the approximation ratio of the method. In particular, by exploiting the favorite structure of the quadratic SDP relaxation, we propose an alternating direction method for solving such a problem and show that the method is globally convergent without any assumption. Some preliminary numerical results are reported which show the effectiveness of the method proposed in this paper.     

On the solution of the integro-differential equation with an integral boundary condition

In this paper, we present an existence of solution for a functional integro-differential equation with an integral boundary condition arising in chemical engineering, underground water flow and population dynamics, and other field of physics and mathematical chemistry. By using the techniques of noncompactness measures, we employ the basic fixed point theorems such as Darbo’s theorem to obtain the mentioned aim in Banach algebra. Then this paper presents a powerful numerical approach based on Sinc approximation to solve the equation. Then convergence of this technique is discussed by preparing a theorem which shows exponential type convergence rate and guarantees the applicability of that. Finally, some numerical examples are given to confirm efficiency and accuracy of the numerical scheme.     

Classical solution of a problem with an integral condition for the one-dimensional wave equation

We find a closed-form classical solution of the homogeneous wave equation with Cauchy conditions, a boundary condition on the lateral boundary, and a nonlocal integral condition involving the values of the solution at interior points of the domain. A classical solution is understood as a function that is defined everywhere in the closure of the domain and has all classical derivatives occurring in the equation and conditions of the problem. The derivatives are defined via the limit values of finite-difference ratios of the function and corresponding increments of the arguments.     

On damped Kirchhoff equation with variable coefficients

In this work we are concerned with the existence and uniqueness of strong global solutions and exponential decay of the total energy for an initial-boundary value problem associated with the Kirchhoff equation with variable coefficients on the action of a nonlinear internal damping.     

Periodic solutions of an iterative functional differential equation with variable coefficients

In this paper, we use Krasnoselskii's fixed point theorem to study the existence and uniqueness of periodic solutions of an iterative functional differential equation Copyright © 2016 John Wiley & Sons, Ltd.     

A meshless method for numerical solution of the one-dimensional wave equation with an integral condition using radial basis functions

The hyperbolic partial differential equation with an integral condition arises in many physical phenomena. In this paper, we propose a numerical scheme to solve the one-dimensional hyperbolic equation that combines classical and integral boundary conditions using collocation points and approximating the solution using radial basis functions (RBFs). The results of numerical experiments are presented, and are compared with analytical solution and finite difference method to confirm the validity and applicability of the presented scheme.     

An exact solution for variable coefficients fourth-order wave equation using the Adomian method

This paper presents a novel approach for the analysis of a fourth-order parabolic equation dealing with vibration of beams by using the decomposition method. In this regard, a general approach based on the generalized Fourier series expansion is applied. The obtained analytic solution is simplified in terms of a given set of orthogonal basis functions. The result is compared with the classical modal analysis technique which is widely used in the field of structural dynamics. It is shown that the result of the decomposition method leads to an exact closed-form solution which is equivalent to the result obtained by the modal analysis method. Some examples are given to demonstrate the validity of the present study.