Application of fuzzy differential transform method for solving fuzzy Volterra integral equations

This paper deals with the solutions of fuzzy Volterra integral equations with separable kernel by using fuzzy differential transform method (FDTM). If the equation considered has a solution in terms of the series expansion of known functions, this powerful method catches the exact solution. To this end, we have obtained several new results to solve mentioned problem when FDTM has been applied. In order to show this capability and robustness, some fuzzy Volterra integral equations are solved in detail as numerical examples.     

Solvability of the boundary-value problem for a variable-order differential equation on a geometric graph

The solvability of the boundary-value problem for a string-beam model is studied. The model is described by an equation of orders 2 or 4 on dirrerent edges of an arbitrary graph. Criteria for the problem to be degenerate and nondegenerate are obtained; in particular, it is proved that the nondegeneracy of the problem is equivalent to the maximum principle.     

Differential transform method for solving Volterra integral equation with separable kernels

In this paper, Volterra integral equations with separable kerenels are solved using the differential transform method. The approximate solution of this equation is calculated in the form of a series with easily computable terms. Exact solutions of linear and nonlinear integral equations have been investigated and the results illustrate the reliability and the performance of the differential transform method.     

Riccati方程的可积性条件

得到了R iccati方程的一些可积的充分条件及其在这些条件下的通积分,此结果包含了已有文献中的一些结论,最后给出定理应用的例子.     

On a mixed problem with integral conditions for one-dimensional parabolic equation

We consider an initial-boundary value problem for a one-dimensional parabolic equation with nonlocal boundary conditions. These nonlocal conditions are given in terms of integrals. Based on solution of the Dirichlet problem for the parabolic equation, we constructively establish the well-posedness for the nonlocal problem.     

带非局部积分项常微分方程的讨论及其应用

研究带非局部积分项的二阶线性常微分方程及其在金融保险上的应用.首先讨论带非局部积分项的二阶常微分方程解的存在唯一性,通过变量代换和累次积分交换积分顺序将非局部项简化,将方程化为方程组,然后完成了对方程组解的存在唯一性的证明.接着分析了带非局部项的二阶常微分方程解的结构,给出了方程解的形式.最后通过推导,指出带非局部项的线性常微分方程在保险公司的破产概率研究中的应用,重点放在二阶方程的应用上,并且在某一特定情况下,举出了一个可以给出解析解的例子.     

The method of integral equation for scattering problem with a mixed crack

We consider the scattering of an electromagnetic time‐harmonic plane wave by an infinite cylinder having a mixed open crack (or arc) in R² as the cross section. The crack is made up of two parts, and one of the two parts is (possibly) coated by a material with surface impedance λ. We transform the scattering problem into a system of boundary integral equations by adopting a potential approach, and establish the existence and uniqueness of a weak solution to the system by the Fredholm theory. Copyright © 2011 John Wiley & Sons, Ltd.     

Application of the discrete Fourier transform to solving the Cauchy integral equation

The uniquely solvable system of the Cauchy integral equation of the first kind and index 1 and an additional integral condition is treated. Such a system arises, for example, when solving the skew derivative problem for the Laplace equation outside an open arc in a plane. This problem models the electric current from a thin electrode in a semiconductor film placed in a magnetic field. A fast and accurate numerical method based on the discrete Fourier transform is proposed. Some computational tests are given. It is shown that the convergence is close to exponential.     

Superconvergence of a discontinuous finite element method for a nonlinear ordinary differential equation

In this paper, n-degree discontinuous finite element method with interpolated coefficients for an initial value problem of nonlinear ordinary differential equation is introduced and analyzed. By using the finite element projection for an auxiliary linear problem as comparison function, an optimal superconvergence , at (n + 1)-order characteristic points in each element respectively is proved. Finally the theoretic results are tested by a numerical example.     

A fast adaptive spectral graph wavelet method for the viscous Burgers' equation on a star-shaped connected graph

Water wave propagation in an open channel network can be described by the viscous Burgers' equation on the corresponding connected graph, possibly with small viscosity. In this paper, we propose a fast adaptive spectral graph wavelet method for the numerical solution of the viscous Burgers' equation on a star-shaped connected graph. The vital feature of spectral graph wavelets is that they can be constructed on any complex network using the graph Laplacian. The essence of the method is that the same operator can be used for the construction of the spectral graph wavelet and the approximation of the differential operator involved in the Burgers' equation. In this paper, two test problems are considered with homogeneous Dirichlet boundary condition. The numerical results show that the method accurately captures the evolution of the localized patterns at all the scales, and the adaptive node arrangement is accordingly obtained. The convergence of the given method is verified, and efficiency is shown using CPU time.     

A finite difference method for fractional partial differential equation

An implicit unconditional stable difference scheme is presented for a kind of linear space–time fractional convection–diffusion equation. The equation is obtained from the classical integer order convection–diffusion equations with fractional order derivatives for both space and time. First-order consistency, unconditional stability, and first-order convergence of the method are proven using a novel shifted version of the classical Grünwald finite difference approximation for the fractional derivatives. A numerical example with known exact solution is also presented, and the behavior of the error is examined to verify the order of convergence.     

On the finite integral transform method for exact bending solutions of fully clamped orthotropic rectangular thin plates

Exact bending solutions of fully clamped orthotropic rectangular thin plates subjected to arbitrary loads are derived using the finite integral transform method. In the proposed mathematical method one does not need to predetermine the deformation function because only the basic governing equations of the classical plate theory for orthotropic plates are used in the procedure. Therefore, unlike conventional semi-inverse methods, it serves as a completely rational and accurate model in plate bending analysis. The applicability of the method is extensive, and it can handle plates with different loadings in a uniform procedure, which is simpler than previous methods. Numerical results are presented to demonstrate the validity and accuracy of the approach as compared with those previously reported in the bibliography.     

非结构网格下2D Riesz分数阶方程的Galerkin有限元方法

讨论了2D Riesz分数阶扩散方程的Galerkin有限元方法.基于非结构网格,采用Lagrange线性分片多项式作为基函数,详细描述了分数阶扩散方程的有限元实现.与现有方法相比,该方法有效地降低了计算成本,提高了刚度矩阵的精度.最后,数值算例验证了所提方法的有效性.     

Weak Galerkin finite element methods for a fourth order parabolic equation

This paper is devoted to a newly developed weak Galerkin finite element method with the stabilization term for a linear fourth order parabolic equation, where weakly defined Laplacian operator over discontinuous functions is introduced. Priori estimates are developed and analyzed in L² and an H² type norm for both semi‐discrete and fully discrete schemes. And finally, numerical examples are provided to confirm the theoretical results.     

Partial differential integral equation model for pricing American option under multi state regime switching with jumps

In this paper, we consider a two dimensional partial differential integral equation (PDIE) model for pricing American option. A nonlinear rationality parameter function for two asset problems is introduced to deal with the free boundary. The rationality parameter function is added in the PDIEs used for pricing American option problems under multi-state regime switching with jumps. The resulting two dimensional nonlinear system of PDIE is then numerically solved. Based on real poles rational approximation, a strongly stable highly efficient and reliable method is developed to solve such complicated systems of PIDEs. The method is build in a predictor corrector style which makes it linearly implicit, therefore, avoids solving nonlinear systems of equations at each time step in all regimes. The method is seen to maintain the stability and convergence for large jump sizes and high volatility in each regime. The impact of regime switching on option prices corresponding to different values interest rate, volatility, and rationality parameter is computed, illustrated by graphs and given in the tables. Convergence results in each regime are presented and time evolution graphs are given to show the effectiveness and reliability of the method.     

Analysis of the physical Laplacian and the heat flow on a locally finite graph

We study the physical Laplacian and the corresponding heat flow on an infinite, locally finite graph with possibly unbounded valence.     

The Hicks property for a variational problem on a graph

In this paper we use the Hicks property for a variational problem on a graph. For an elastic system defined on a graph we state a well-posed problem which implies the definition and the study of the influence function.