Eulerian–Lagrangian localized adjoint methods for reactive transport with biodegradation

The microbial degradation of organic contaminants in the subsurface holds significant potential as a mechanism for in-situ remediation strategies. The mathematical models that describe contaminant transport with biodegradation involve a set of advective–diffusive–reactive transport equations. These equations are coupled through the nonlinear reaction terms, which may involve reactions with all of the species and are themselves coupled to growth equations for the subsurface bacterial populations. In this article, we develop Eulerian–Lagrangian localized adjoint methods (ELLAM) to solve these transport equations. ELLAM are formulated to systematically adapt to the changing features of governing partial differential equations. The relative importance of retardation, advection, diffusion, and reaction is directly incorporated into the numerical method by judicious choice of the test functions that appear in the weak form of the governing equation. Different ELLAM schemes for linear variable–coefficient advective–diffusive–reactive transport equations are developed based on different operator splittings. Specific linearization techniques are discussed and are combined with the ELLAM schemes to solve the nonlinear, multispecies transport equations. © 1995 John Wiley & Sons, Inc.     

Application of Haar wavelet method to eigenvalue problems of high order differential equations

In this work, we present a computational method for solving eigenvalue problems of high-order ordinary differential equations which based on the use of Haar wavelets. The variable and their derivatives in the governing equations are represented by Haar function and their integral. The first transform the spectral coefficients into the nodal variable values. The second, solve the obtained system of algebraic equation. The efficiency of the method is demonstrated by four numerical examples.     

Laplace transforms and American options

Laplace transform methods are used to study the valuation of American call and put options with constant dividend yield, and to derive integral equations giving the location of the optimal exercise boundary. In each case studied, the main result of this paper is a nonlinear Fredholm-type integral equation for the location of the free boundary. The equations differ depending on whether the dividend yield is less than or exceeds the risk-free rate. These integral equations contain a transform variable, so the solution of the equations would involve finding the free boundary that satisfies the equations for all values of this transform variable. Expressions are also given for the transform of the value of the option in terms of this free boundary.     

A new analytical technique to solve Volterra's integral equations

The purpose of the present paper is to show that the well‐known homotopy analysis method for solving ordinary and partial differential equations can be applied to solve linear and nonlinear integral equations of Volterra's type with high accuracy as well. Comparison of the present method with Adomian decomposition method (ADM), a well‐known method to solve integral equations, reveals that the ADM is only especial case of the present method. Furthermore, some illustrating examples such as linear, nonlinear and singular integral equations of Volterra's type are given to show high efficiency with reliable accuracy of the method. Copyright © 2011 John Wiley & Sons, Ltd.     

On the numerical solution of stochastic quadratic integral equations via operational matrix method

In this paper, stochastic operational matrix of integration based on delta functions is applied to obtain the numerical solution of linear and nonlinear stochastic quadratic integral equations (SQIEs) that appear in modelling of many real problems. An important advantage of this method is that it dose not need any integration to compute the constant coefficients. Also, this method can be utilized to solve both linear and nonlinear problems. By using stochastic operational matrix of integration together collocation points, solving linear and nonlinear SQIEs converts to solve a nonlinear system of algebraic equations, which can be solved by using Newton's numerical method. Moreover, the error analysis is established by using some theorems. Also, it is proved that the rate of convergence of the suggested method is O(h²). Finally, this method is applied to solve some illustrative examples including linear and nonlinear SQIEs. Numerical experiments confirm the good accuracy and efficiency of the proposed method.     

Computational method based on Bernstein operational matrices for nonlinear Volterra-Fredholm-Hammerstein integral equations

In this paper, we present a method to solve nonlinear Volterra-Fredholm-Hammerstein integral equations in terms of Bernstein polynomials. Properties of these polynomials and operational matrix of integration together with the product operational matrix are first presented. These properties are then utilized to transform the integral equation to a matrix equation which corresponds to a system of nonlinear algebraic equations with unknown Bernstein coefficients. The method is computationally very simple and attractive and numerical examples illustrate the efficiency and accuracy of the method.     

Interval solution of nonlinear equations using linear programming

A new computational test is proposed for nonexistence of a solution to a system of nonlinear equations in a convex polyhedral regionX. The basic idea proposed here is to formulate a linear programming problem whose feasible region contains all solutions inX. Therefore, if the feasible region is empty (which can be easily checked by Phase I of the simplex method), then the system of nonlinear equations has no solution inX. The linear programming problem is formulated by surrounding the component nonlinear functions by rectangles using interval extensions. This test is much more powerful than the conventional test if the system of nonlinear equations consists of many linear terms and a relatively small number of nonlinear terms. By introducing the proposed test to interval analysis, all solutions of nonlinear equations can be found very efficently. This work was partially supported by the Japanese Ministry of Education.     

New technique to avoid “noise terms” on the solutions of inhomogeneous differential equations by using Adomian decomposition method

In this paper we present a new technique to get the solutions of inhomogeneous differential equations using Adomian decomposition method (ADM) without noise terms. We construct an appropriate differential equations for the inhomogeneity function which must be contains the integral variable, and convert all of these differential equations (original differential equation and the constructed differential equations) to augmented system of first-order differential equations. The ADM is using to solve the augmented system and the initial conditions are taken as initial approximations. Generally, the closed form of the exact solution or its expansion is obtained without any noise terms. Several differential equations will be tested to confirm the newly developed technique.     

A new analytical technique to solve Fredholm’s integral equations

This paper shows that the homotopy analysis method, the well-known method to solve ODEs and PDEs, can be applied as well as to solve linear and nonlinear integral equations with high accuracy. Comparison of the present method with Adomian decomposition method (ADM), which is well-known in solving integral equations, reveals that the ADM is only special case of the present method. Also, some linear and nonlinear examples are presented to show high efficiency and illustrate the steps of the problem resolution.     

B-Spline collocation method for linear and nonlinear Fredholm and Volterra integro-differential equations

A numerical method based on quintic B-spline has been developed to solve the linear and nonlinear Fredholm and Volterra integro-differential equations up to order 4. The solution and its derivatives are collocated by quintic B-spline and then the integral equation is approximated by the 4-points Gauss–Turán quadrature formula with respect to the weight function Legendre. The error analysis of proposed numerical method is studied theoretically. Numerical results are given to illustrate the efficiency of the proposed method which shows that our method can be applied for large values of N. The results are compared with the results obtained by other methods which show that our method is accurate.     

Semiconjugate factorizations of higher order linear difference equations in rings

We use a new nonlinear method to study linear difference equations with variable coefficients in a non-trivial ring R. If the homogeneous part of the linear equation has a solution in the unit group of a ring with identity (a unitary solution), then we show that the equation decomposes into two linear equations of lower orders. This decomposition, known as a semiconjugate factorization in the nonlinear theory, is based on sequences of ratios of consecutive terms of a unitary solution. Such sequences, which may be called eigensequences, are well suited to variable coefficients; for instance, they provide a natural context for the expression of the Poincaré–Perron theorem. As applications, we obtain new results for linear difference equations with periodic coefficients and for linear recurrences in rings of functions (e.g. the recurrence for the modified Bessel functions).     

Numerical solution of volterra functional integral equation by using cubic B‐spline scaling functions

In this article, we consider a class of nonlinear functional integral equations which has rather general form and contains a lot of particular cases such as functional equations and nonlinear integral equations of Volterra type. We use a combination of a fixed point method and cubic semiorthogonal B‐spline scaling functions to solve the integral equation numerically. We provide an error analysis for the method which shows that the approximate solution converges to the exact solution. Some numerical results for several test problems are given to confirm the accuracy and the ease of implementation of the method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 699–722, 2014     

Global polynomial approximation for Symm's equation on polygons

Summary. To solve 1D linear integral equations on bounded intervals with nonsmooth input functions and solutions, we have recently proposed a quite general procedure, that is essentially based on the introduction of a nonlinear smoothing change of variable into the integral equation and on the approximation of the transformed solution by global algebraic polynomials. In particular, the new procedure has been applied to weakly singular equations of the second kind and to solve the generalized air foil equation for an airfoil with a flap. In these cases we have obtained arbitrarily high orders of convergence through the solution of very-well conditioned linear systems. In this paper, to enlarge the domain of applicability of our technique, we show how the above procedure can be successfully used also to solve the classical Symm's equation on a piecewise smooth curve. The collocation method we propose, applied to the transformed equation and based on Chebyshev polynomials of the first kind, has shown to be stable and convergent. A comparison with some recent numerical methods using splines or trigonometric polynomials shows that our method is highly competitive. Received October 1, 1998 / Revised version received September 27, 1999 / Published online June 21, 2000     

波纹壳的格林函数方法

应用轴对称旋转扁壳的基本方程,研究了在任意载荷作用下具有型面锥度的浅波纹壳的非线性弯曲问题·　采用格林函数方法,将扁壳的非线性微分方程组化为非线性积分方程组·　再使用展开法求出格林函数,即将格林函数展成特征函数的级数形式,积分方程就成为具有退化核的形式,从而容易得到非线性代数方程组·　应用牛顿法求解非线性代数方程组时,为了保证迭代的收敛性,选取位移作为控制参数,逐步增加位移,求得相应的载荷·　在算例中,研究了具有球面度的浅波纹壳的弹性特征·　结果表明,由于型面锥度的引入,特征曲线发生显著变化,随着荷载的增加,将出现类似扁球壳的总体失稳现象·　本文的解答符合实验结果·     

A note on a conjecture for the critical curve of a weakly coupled system of semilinear wave equations with scale-invariant lower order terms

In this note, two blow-up results are proved for a weakly coupled system of semilinear wave equations with distinct scale-invariant lower order terms both in the subcritical case and in the critical case when the damping and the mass terms make both equations in some sense “wave-like.” In the proof of the subcritical case, an iteration argument is used. This approach is based on a coupled system of nonlinear ordinary integral inequalities and lower bound estimates for the spatial integral of the nonlinearities. In the critical case, we employ a test function-type method that has been developed recently by Ikeda-Sobajima-Wakasa and relies strongly on a family of certain self-similar solutions of the adjoint linear equation. Therefore, as critical curve in the p−q plane of the exponents of the power nonlinearities for this weakly coupled system, we conjecture a shift of the critical curve for the corresponding weakly coupled system of semilinear wave equations.     

Mixed methods of nonlinear second-order elliptic problems in three variables

Mixed finite element methods for treating the Dirichlet problem for fully nonlinear second-order elliptic operators in divergence form are extended to cover the three-dimensional case. Existence and uniqueness of the approximation are proved, and optimal error estimates in L² are demonstrated for both the scalar and vector functions approximated by the method. Error estimates for the pressure variable are also derived in L^q; the result is optimal in order for 2 ≤ q ≤ 6 and less than optimal for 6 < q ≤ + ∞. Newton's method can be used to solve the nonlinear algebraic equations. © 1996 John Wiley & Sons, Inc.     

变质量非线性非完整系统的Gibbs-Appell方程

本文首先将Gibbs-Appell方程推广到最一般的变质量非完整系统.得到变质量非线性非完整系统在广义坐标、准坐标下的Gibbs-Appell方程和积分变分原理,最后给出一个例子.     

A Taylor polynomial approach for solving the most general linear Fredholm integro‐differential‐difference equations

In this study, a matrix method is developed to solve approximately the most general higher order linear Fredholm integro‐differential‐difference equations with variable coefficients under the mixed conditions in terms of Taylor polynomials. This technique reduces the problem into the linear algebraic system. The method is valid for any combination of differential, difference and integral equations. An initial value problem and a boundary value problem are also presented to illustrate the accuracy and efficiency of the method. Copyright © 2012 John Wiley & Sons, Ltd.     

Natural frequencies of thin rectangular plates using homotopy-perturbation method

The homotopy perturbation method (HPM) was developed to search for asymptotic solutions of nonlinear problems involving parabolic partial differential equations with variable coefficients. This paper illustrates that HPM be easily adapted to solve parabolic partial differential equations with constant coefficients. Natural frequencies of a rectangular plate of uniform thickness, simply-supported on all sides, are obtained with minimum amount of computation. The solution is shown to converge rapidly to a combination of sine and cosine functions. Truncating the series solution by using only the first three terms of the sine and cosine functions as compared to the exact solution results in an absolute error not exceeding 2 × 10⁻⁴ and 9×10⁻⁴ for the trigonometric functions respectively. HPM is then applied to solve the nonlinear problem of a rectangular plate of variable thickness. A direct expression for the eigenvalues (natural frequencies) of the rectangular plate is obtained as compared to determining its eigenvalues by solving the characteristic equation using the conventional method. Comparison of results for the frequency parameter with existing literature show that HPM is highly efficient and accurate. Natural frequencies of a simply-supported guitar soundboard were obtained using an equivalent rectangular plate with the same boundary condition.     

An Eulerian–Lagrangian mixed discrete least squares meshfree method for incompressible multiphase flow problems

Mixed discrete least squares meshfree (MDLSM) method has been developed as a truly meshfree method and successfully used to solve single-phase flow problems. In the MDLSM, a residual functional is minimized in terms of the nodal unknown parameters leading to a set of positive-definite system of algebraic equations. The functional is defined using a least square summation of the residual of the governing partial differential equations and its boundary conditions at all nodal points discretizing the computational domain. Unlike the discrete least squares meshfree (DLSM) which uses an irreducible form of the governing equations, the MDLSM uses a mixed form of the original governing equations allowing for direct calculation of the gradients leading to more accurate computational results. In this study, an Eulerian–Lagrangian MDLSM method is proposed to solve incompressible multiphase flow problems. In the Eulerian step, the MDLSM method is used to solve the governing phase averaged Navier–Stokes equations discretized at fixed nodal points to get the velocity and pressure fields. A Lagrangian based approach is then used to track different flow phases indexed by a set of marker points. The velocities of marker points are calculated by interpolating the velocity of fixed nodal points using a kernel approximation, which are then used to move the marker points as Lagrangian particles to track phases. To avoid unphysical clustering and dispersing of the marker points, as a common drawback of Lagrangian point tracking methods, a new approach is proposed to smooth the distribution of marker points. The hybrid Eulerian and Lagrangian characteristics of the approach used here provides clear advantages for the proposed method. Since the nodal points are static on the Eulerian step, the time-consuming moving least squares (MLS) approximation is implemented only once making the proposed method more efficient than corresponding fully Lagrangian methods. Furthermore, phases can be simply tracked using the Lagrangian phase tracking procedure. Efficiency of the proposed MDLSM multiphase method is evaluated using several benchmark problems and the results are presented and discussed. The results verify the efficiency and accuracy of the proposed method for solving multiphase flow problems.