Numerical study of Fisher's equation by wavelet Galerkin method

Fisher's equation, which describes the logistic growth–diffusion process and occurs in many biological and chemical processes, has been studied numerically by the wavelet Galerkin method. Wavelets are functions which can provide local finer details. The solution of Fisher's equation has a compact support property and therefore Daubechies' compactly supported wavelet basis has been used in this study. The results obtained by the present method are highly encouraging and can be computed for a large value of the linear growth rate.     

A fully discrete local discontinuous Galerkin method for one-dimensional time-fractional Fisher's equation

In this paper, we consider the local discontinuous Galerkin (LDG) finite element method for one-dimensional time-fractional Fisher's equation, which is obtained from the standard one-dimensional Fisher's equation by replacing the first-order time derivative with a fractional derivative (of order α, with 0<α<1). The proposed LDG is based on the LDG finite element method for space and finite difference method for time. We prove that the method is stable, and the numerical solution converges to the exact one with order O(h^k+1+τ^2?α), where h, τ and k are the space step size, time step size, polynomial degree, respectively. The numerical experiments reveal that the LDG is very effective.     

The dual reciprocity boundary element formulation for nonlinear diffusion problems

This paper presents an extension of the dual reciprocity boundary element method (DRBEM) to deal with nonlinear diffusion problems in which thermal conductivity, specific heat, and density coefficients are all functions of temperature. The DRBEM, recently applied to the solution of problems governed by parabolic and hyperbolic equations, consists in the transformation of the differential equation into an integral equation involving boundary integrals only, the solution of which is achieved by employing a standard boundary element discretization coupled with a two-level finite difference time integration scheme. Contrary to previous formulations for the diffusion equation, the dual reciprocity BEM utilizes the well-known fundamental solution to Laplace's equation, which is space-dependent only. This avoids complex time integrations that normally appear in formulations employing time-dependent fundamental solutions, and permits accurate numerical solutions to be obtained in an efficient way. For nonlinear problems, the integral of conductivity is introduced as a new variable to obtain a linear diffusion equation in the Kirchhoff transform space. This equation involves a modified time variable which is itself a function of position. The problem is solved in an iterative way by using an efficient Newton-Raphson technique which is shown to be rapidly convergent.     

On combination of first order Strang's splitting and additive splitting for the solution of multi-dimensional convection diffusion problem

In this article we present a new splitting approach for the numerical solution of the multi-dimensional convection diffusion equations. The method combines additive and multiplicative splitting. In particular the method combines first order Strang's splitting, multiplicative splitting defined for splitting the convection and diffusion equation, and additive splitting defined in accordance with the spatial variables. The method not only reduces the linear (or nonlinear) original problem into a series of one-dimensional and one physical operator linear problems, but also enables us to compute these one-dimensional problems using parallel processors. The accuracy and stability of the new algorithm are investigated through the solution of different multi-dimensional convection diffusion model problems with scalar coefficients.     

A numerical algorithm for computation modelling of 3D nonlinear wave equations based on exponential modified cubic B-spline differential quadrature method

In this paper, the authors proposed a method based on exponential modified cubic B-spline differential quadrature method (Expo-MCB-DQM) for the numerical simulation of three dimensional (3D) nonlinear wave equations subject to appropriate initial and boundary conditions. This work extends the idea of Tamsir et al. [An algorithm based on exponential modified cubic B-spline differential quadrature method for nonlinear Burgers’ equation, Appl. Math. Comput. 290 (2016), pp. 111–124] for 3D nonlinear wave type problems. Expo-MCB-DQM transforms the 3D nonlinear wave equation into a system of ordinary differential equations (ODEs). To solve the resulting system of ODEs, an optimal five stage and fourth-order strong stability preserving Runge–Kutta (SSP-RK54) scheme is used. Stability analysis of the proposed method is also discussed and found that the method is conditionally stable. Four test problems are considered in order to demonstrate the accuracy and efficiency of the algorithm.     

A fourth-order B-spline collocation method and its error analysis for Bratu-type and Lane–Emden problems

Recently, Caglar et al. [B-spline method for solving Bratu's problem, Int. J. Comput. Math. 87(8) (2010), pp. 1885–1891] proposed a numerical technique based on cubic B-spline for solving a Bratu-type problem. This method provides a second-order convergent approximation to the solution of the problem. In this paper, we develop a high-order numerical method based on quartic B-spline collocation approach for the Bratu-type and Lane–Emden problems. The error analysis of the quartic B-spline interpolation is carried out. Some numerical examples are provided to demonstrate the efficiency and applicability of the method and to verify its rate of convergence. The numerical results are compared with exact solutions and a numerical method based on cubic B-spline approach. Comparison reveals that our method produces more accurate results than the method proposed by Caglar et al. [B-spline method for solving Bratu's problem, Int. J. Comput. Math. 87(8) (2010), pp. 1885–1891].     

B-样条函数极小曲面造型

极小曲面在建筑、航空、轮船制造等领域有着重要应用,但由于极小曲面表示复杂,给实际应用带来了很大的困难.研究了具有给定边界的极小曲面的B-样条函数曲面逼近.基于非线性约束优化方法和有限单元方法,求极小曲面方程的近似解.在算法中使用数值延拓方法,使非线性问题的初值选择问题自动化,同时,使用一个简单的线性化策略对非线性问题进行线性化.给出了几个数值结果.     

B-spline method for solving Bratu's problem

In this paper, we propose a B-spline method for solving the one-dimensional Bratu's problem. The numerical approximations to the exact solution are computed and then compared with other existing methods. The effectiveness and accuracy of the B-spline method is verified for different values of the parameter, below its critical value, where two solutions occur.     

Diffusion and dispersion errors of the flux corrected upwind method

The inviscid form of Burger's equation is solved numerically using the flux corrected upwind method. A nonlinear Hirt stability analysis is carried out. It is shown that the flux limiting process is responsible for modifying the advection speed in the shock region to reduce the dispersion errors. The linear and the nonlinear diffusion coefficients are of the same sign i.e. diffusion errors are dominant. The flux limiter preserves the conservation property.     

Design of heterogeneous turbine blade

Constantly rising operating pressure and temperature in turbine drivers push the material capabilities of turbine blades to the limit. The recent development of heterogeneous objects by layered manufacturing offers new potentials for the turbine blades. In heterogeneous turbine blades, multiple materials can be synthesized to provide better properties than any single material. A critical task of such synthesis in turbine blade design is an effective design method that allows a designer to design geometry and material composition simultaneously.This paper presents a new approach for turbine blade design, which ties B-spline representation of a turbine blade to a physics (diffusion) process. In this approach, designers can control both geometry and material composition. Meanwhile, material properties are directly conceivable to the designers during the design process. The designer's role is enhanced from merely interpreting the optimization result to explicitly controlling both material composition and geometry according to the acquired experience (material property constraints).The mathematical formulation of the approach includes three steps: using B-spline to represent the turbine blade, using diffusion equation to generate material composition variation, using finite element method to solve the constrained diffusion equation. The implementation and examples are presented to validate the effectiveness of this approach for heterogeneous turbine blade design.     

Diffusive approximation of fisher's equation

We show that Fisher's Equation is the relaxation limit of a discrete kinetic model with two speeds, in which the collision term contains a source. The analysis can be directly applicable to obtain a numerical approximation to Fisher's Equation in terms of the relaxing system.     

Generalized Laguerre spectral method for Fisher's equation on a semi-infinite interval

In this paper, we propose a generalized Laguerre spectral method for Fisher's-type equation with inhomogeneous boundary conditions on a semi-infinite interval. By reformulating the equation with suitable functional transform, it is shown that the generalized Laguerre approximations are convergent on a semi-infinite interval with spectral accuracy. An efficient and accurate algorithm based on the generalized Laguerre approximations to the transformed equation is developed and implemented. Numerical results show the efficiency of this approach and coincide well with theoretical analysis.     

Asymptotic output tracking for a class of semilinear parabolic equations: A semianalytical approach

This paper addresses the problem of asymptotic output tracking of a class of semilinear parabolic equations with pointwise in‐domain actuation. First, the assessment of the well‐posedness of the considered systems is performed, and then, the stability of boundary controlled systems is analyzed via Chaffee‐Infante equation and Fisher's equation. The application of the zero dynamics inverse design results in a dynamic control scheme that is implemented by using the technique of trajectory planning for flat systems and the Adomian decomposition method. The convergence of the solution of the original systems to that of the corresponding zero dynamics and the convergence of the solution expressed by an Adomian series are also analyzed. Numerical simulations are carried out to illustrate the effectiveness of the developed approach.     

改进的P-M扩散方程在相干斑抑制中的应用

直接基于Perona-Malik扩散方程的滤波算法对于加性噪声非常有效,但是对于乘性噪声（如合成孔径雷达（SAR）图像相干斑噪声）收效甚微。提出了一种基于改进的Perona-Malik扩散方程抑制SAR图像相干斑噪声的新算法。分析对数变化对相干斑噪声的影响,为将P-M扩散方程应用于相干斑噪声抑制奠定了理论基础;通过P-M扩散和稳健统计学的联系,建立了基于Biweight Estimator误差模型的扩散系数;同时利用非线性衰减技术对梯度阈值的选择改进。实验表明,该方法不仅有效抑制了SAR图像相干斑噪声,较好地保持了细节和边缘信息,而且视觉效果比较好。     

Nonlinear registration using variational principle for mutual information

We extend the multimodal image registration method described in Alexander and Summers [Fast registration algorithm using a variational principle for mutual information, Proc. SPIE Int. Soc. Opt. Eng. 5032 (2003) 1053–1063] to nonlinear registration. A variational principle maximizing mutual information leads to an Euler–Lagrange (EL) equation for the displacement field, represented here in a basis of cubic B-spline functions. A cost function is constructed from the sum of squares of the residuals of the EL equation at a subset of pixels where the magnitude of the spatial gradient of intensity exceeds a user-chosen threshold. The unknown coefficients in the displacement field representation are evaluated using a Levenberg–Marquardt minimization procedure. The proposed method was successfully applied to several image pairs of the same and different modalities, and an artificially constructed series of images containing nonlinear distortions and noise.     

Computationally efficient simultaneous policy update algorithm for nonlinear H∞ state feedback control with Galerkin's method

The main bottleneck for the application of H_∞ control theory on practical nonlinear systems is the need to solve the Hamilton–Jacobi–Isaacs (HJI) equation. The HJI equation is a nonlinear partial differential equation (PDE) that has proven to be impossible to solve analytically, even the approximate solution is still difficult to obtain. In this paper, we propose a simultaneous policy update algorithm (SPUA), in which the nonlinear HJI equation is solved by iteratively solving a sequence of Lyapunov function equations that are linear PDEs. By constructing a fixed point equation, the convergence of the SPUA is established rigorously by proving that it is essentially a Newton's iteration method for finding the fixed point. Subsequently, a computationally efficient SPUA (CESPUA) based on Galerkin's method, is developed to solve Lyapunov function equations in each iterative step of SPUA. The CESPUA is simple for implementation because only one iterative loop is included. Through the simulation studies on three examples, the results demonstrate that the proposed CESPUA is valid and efficient. Copyright © 2012 John Wiley & Sons, Ltd.     

求解对流扩散方程的混合三次B-样条配点法

通过对三次B-样条和三次三角B-样条基函数引入权因子[ω],给出了对流扩散方程的混合三次B-样条配点法。对对流扩散方程空间离散采用混合三次B-样条配点法和时间离散采用向前有限差分,引入参数[θ],建立差分格式。对差分格式的稳定性进行分析,得到稳定性条件。数值实验表明所构造方法的有效性,并且适当调整权因子[ω]和参数[θ]的值,可提高计算的精度。     

Collocation and finite difference-collocation methods for the solution of nonlinear Klein-Gordon equation

Two numerical techniques based on the finite difference and collocation methods are presented for the solution of nonlinear Klein-Gordon equation. The operational matrix of derivative for the cubic B-spline scaling functions is presented and is utilized to reduce the solution of nonlinear Klein-Gordon equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the new techniques.     

Series solution of nonlinear differential equations by a novel extension of the Laplace transform method

A technique for extending the Laplace transform method to solve nonlinear differential equations is presented. By developing several theorems, which incorporate the Adomian polynomials, the Laplace transformation of nonlinear expressions is made possible. A number of well-known nonlinear equations including the Riccati equation, Clairaut's equation, the Blasius equation and several other ones involving nonlinearities of various types such as exponential and sinusoidal are solved for illustration. The proposed approach is analytical, accurate, and free of integration.     

Numerical solution of homogeneous Smoluchowski's coagulation equation

Smoluchowski's equation is widely applied to describe the time evolution of the cluster-size distribution during aggregation processes. Analytical solutions for this equation, however, are known only for a very limited number of kernels. Therefore, numerical methods have to be used to describe the time evolution of the cluster-size distribution. A numerical technique is presented for the solution of the homogeneous Smoluchowski's coagulation equation with constant kernel. In this paper, we use Taylor polynomials and radial basis functions together to solve the equation. This method converts Smoluchowski's equation to a system of nonlinear equations that can be solved for unknown parameters. A numerical example with known solution is included to demonstrate the validity and applicability of the technique.