Showing papers in "American Journal of Computational Mathematics in 2011"

Application of Multi-Step Differential Transform Method on Flow of a Second-Grade Fluid over a Stretching or Shrinking Sheet

Abstract: In this study, a reliable algorithm to develop approximate solutions for the problem of fluid flow over a stretching or shrinking sheet is proposed. It is depicted that the differential transform method (DTM) solutions are only valid for small values of the independent variable. The DTM solutions diverge for some differential equations that extremely have nonlinear behaviors or have boundary-conditions at infinity. For this reason the governing boundary-layer equations are solved by the Multi-step Differential Transform Method (MDTM). The main advantage of this method is that it can be applied directly to nonlinear differential equations without requiring linearization, discretization, or perturbation. It is a semi analytical-numerical technique that formulizes Taylor series in a very different manner. By applying the MDTM the interval of convergence for the series solution is increased. The MDTM is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions for systems of differential equations. It is predicted that the MDTM can be applied to a wide range of engineering applications.

On the Location of Zeros of Polynomials

Abstract: In this paper, we prove some extensions and generalizations of the classical Enestrom-Kakeya theorem.

A Fourth Order Improved Numerical Scheme for the Generalized Burgers—Huxley Equation

Abstract: A fourth order finite-difference scheme in a two-time level recurrence relation is proposed for the numerical solution of the generalized Burgers--Huxley equation. The resulting nonlinear system, which is analysed for stability, is solved using an improved predictor-corrector method. The efficiency of the proposed method is tested to the kink wave using both appropriate boundary values and conditions. The results arising from the experiments are compared with the relevant ones known in the available bibliography.

Haar Wavelet Quasilinearization Approach for Solving Nonlinear Boundary Value Problems

Abstract: Objective of our paper is to present the Haar wavelet based solutions of boundary value problems by Haar collocation method and utilizing Quasilinearization technique to resolve quadratic nonlinearity in y. More accurate solutions are obtained by wavelet decomposition in the form of a multiresolution analysis of the function which represents solution of boundary value problems. Through this analysis, solutions are found on the coarse grid points and refined towards higher accuracy by increasing the level of the Haar wavelets. A distinctive feature of the proposed method is its simplicity and applicability for a variety of boundary conditions. Numerical tests are performed to check the applicability and efficiency. C++ program is developed to find the wavelet solution.

The Odd-Point Ternary Approximating Schemes

Abstract: We present a general formula to generate the family of odd-point ternary approximating subdivision schemes with a shape parameter for describing curves. The influence of parameter to the limit curves and the sufficient conditions of the continuities from C0 to C5 of 3- and 5-point schemes are discussed. Our family of 3-point and 5-point ternary schemes has higher order of derivative continuity than the family of 3-point and 5-point schemes presented by [Jian-ao Lian, On a-ary subdivision for curve design: II. 3-point and 5-point interpolatory schemes, Applications and Applied Mathematics: An International Journal, 3(2), 2008, 176-187]. Moreover, a 3-point ternary cubic B-spline is special case of our family of 3-point ternary scheme. The visual quality of schemes with examples is also demonstrated.

Computing the Moore-Penrose Inverse of a Matrix through Symmetric Rank-One Updates

Abstract: This paper presents a recursive procedure to compute the Moore-Penrose inverse of a matrix A. The method is based on the expression for the Moore-Penrose inverse of rank-one modified matrix. The computational complexity of the method is analyzed and a numerical example is included. A variant of the algorithm with lower computational complexity is also proposed. Both algorithms are tested on randomly generated matrices. Numerical performance confirms our theoretic results.

Chebyshev Approximate Solution to Allocation Problem in Multiple Objective Surveys with Random Costs

Abstract: In this paper, we consider an allocation problem in multivariate surveys as a convex programming problem with non-linear objective functions and a single stochastic cost constraint. The stochastic constraint is converted into an equivalent deterministic one by using chance constrained programming. The resulting multi-objective convex programming problem is then solved by Chebyshev approximation technique. A numerical example is presented to illustrate the computational procedure.

Magneto Hydrodynamic Orthogonal Stagnation Point Flow of a Power-Law Fluid toward a Stretching Surface

Abstract: Steady two dimensional MHD stagnation point flow of a power law fluid over a stretching surface is investigated when the surface is stretched in its own plane with a velocity proportional to the distance from the stagnation point. The fluid impinges on the surface is considered orthogonally. Numerical and analytical solutions are obtained for different cases.

Approximate Solution of the Singular-Perturbation Problem on Chebyshev-Gauss Grid

Abstract: Matrix methods, now-a-days, are playing an important role in solving the real life problems governed by ODEs and/or by PDEs. Many differential models of sciences and engineers for which the existing methodologies do not give reliable results, these methods are solving them competitively. In this work, a matrix methods is presented for approximate solution of the second-order singularly-perturbed delay differential equations. The main characteristic of this technique is that it reduces these problems to those of solving a system of algebraic equations, thus greatly simplifying the problem. The error analysis and convergence for the proposed method is introduced. Finally some experiments and their numerical solutions are given.

An O(k 2 +kh 2 +h 2 ) Accurate Two-level Implicit Cubic Spline Method for One Space Dimensional Quasi-linear Parabolic Equations

Abstract: In this piece of work, using three spatial grid points, we discuss a new two-level implicit cubic spline method of O(k2 + kh2 + h4) for the solution of quasi-linear parabolic equation , 0 0 subject to appropriate initial and Dirichlet boundary conditions, where h > 0, k > 0 are grid sizes in space and time-directions, respectively. The cubic spline approximation produces at each time level a spline function which may be used to obtain the solution at any point in the range of the space variable. The proposed cubic spline method is applicable to parabolic equations having singularity. The stability analysis for diffusion- convection equation shows the unconditionally stable character of the cubic spline method. The numerical tests are performed and comparative results are provided to illustrate the usefulness of the proposed method.

General Solution of Generalized (2+1)–Dimensional Kadomtsev-Petviashvili (KP) Equation by Using the –Expansion Method

Abstract: In this work, the (G,/G)- --expansion method is proposed for constructing more general exact solutions of the (2 + 1)--dimensional Kadomtsev-Petviashvili (KP) equation and its generalized forms. Our work is motivated by the fact that the (G,/G)---expansion method provides not only more general forms of solutions but also periodic and solitary waves. If we set the parameters in the obtained wider set of solutions as special values, then some previously known solutions can be recovered. The method appears to be easier and faster by means of a symbolic computation system.

Conjugate Effects of Radiation and Joule Heating on Magnetohydrodynamic Free Convection Flow along a Sphere with Heat Generation

Abstract: The conjugate effects of radiation and joule heating on magnetohydrodynamic (MHD) free convection flow along a sphere with heat generation have been investigated in this paper. The governing equations are transformed into dimensionless non-similar equations by using set of suitable transformations and solved numerically by the finite difference method along with Newton’s linearization approximation. Attention has been focused on the evaluation of shear stress in terms of local skin friction and rate of heat transfer in terms of local Nusselt number, velocity as well as temperature profiles. Numerical results have been shown graphically for some selected values of parameters set consisting of heat generation parameter Q, radiation parameter Rd, magnetic parameter M, joule heating parameter J and the Prandtl number Pr.

An Efficient Direct Method to Solve the Three Dimensional Poisson’s Equation

Abstract: In this work, the three dimensional Poisson’s equation in Cartesian coordinates with the Dirichlet’s boundary conditions in a cube is solved directly, by extending the method of Hockney. The Poisson equation is approximated by 19-points and 27-points fourth order finite difference approximation schemes and the resulting large algebraic system of linear equations is treated systematically in order to get a block tri-diagonal system. The efficiency of this method is tested for some Poisson’s equations with known analytical solutions and the numerical results obtained show that the method produces accurate results. It is shown that 19-point formula produces comparable results with 27-point formula, though computational efforts are more in 27-point formula.

Quasi-Reversibility Regularization Method for Solving a Backward Heat Conduction Problem

Abstract: Non-standard backward heat conduction problem is ill-posed in the sense that the solution(if it exists) does not depend continuously on the data. In this paper, we propose a regularization strategy-quasi-reversibility method to analysis the stability of the problem. Meanwhile, we investigate the roles of regularization parameter in this method. Numerical result show that our algorithm is effective and stable.

Numerical Solution of Nonlinear Fredholm-Volterra Integtral Equations via Piecewise Constant Function by Collocation Method

Abstract: In this work, we present a computational method for solving nonlinear Fredholm-Volterra integral equations of the second kind which is based on replacement of the unknown function by truncated series of well known Block-Pulse functions (BPfs) expansion. Error analysis is worked out that shows efficiency of the method. Finally, we also give some numerical examples.

Simulation of Human Phonation with Vocal Nodules

Abstract: The geometric and biomechanical properties of the larynx strongly influence voice quality and efficiency. A physical understanding of phonation natures in pathological conditions is important for predictions of how voice disorders can be treated using therapy and rehabilitation. Here, we present a continuum-based numerical model of phonation that considers complex fluid-structure interactions occurring in the airway. This model considers a three-dimensional geometry of vocal folds, muscle contractions, and viscoelastic properties to provide a realistic framework of phonation. The vocal fold motion is coupled to an unsteady compressible respiratory flow, allowing numerical simulations of normal and diseased phonations to derive clear relationships between actual laryngeal structures and model parameters such as muscle activity. As a pilot analysis of diseased phonation, we model vocal nodules, the mass lesions that can appear bilaterally on both sides of the vocal folds. Comparison of simulations with and without the nodules demonstrates how the lesions affect vocal fold motion, consequently restricting voice quality. Furthermore, we found that the minimum lung pressure required for voice production increases as nodules move closer to the center of the vocal fold. Thus, simulations using the developed model may provide essential insight into complex phonation phenomena and further elucidate the etiologic mechanisms of voice disorders.

Computational Optimization of Manufacturing Batch Size and Shipment for an Integrated EPQ Model with Scrap

Abstract: This paper employs mathematical modeling and algebraic approach to derive the optimal manufacturing batch size and number of shipment for a vendor-buyer integrated economic production quantity (EPQ) model with scrap. Unlike the conventional method by using differential calculus to determine replenishment lot size and optimal number of shipments for such an integrated system, this paper proposes a straightforward algebraic approach to replace the use of calculus on the total cost function for solving the optimal production- shipment policies. A simpler form for computing long-run average cost for such a vendor- buyer integrated EPQ problem is also provided.

Solution of the Generalized Abel Integral Equation by Using Almost Bernstein Operational Matrix

Abstract: A direct almost Bernstein operational matrix of integration is used to propose a stable algorithm for numerical inversion of the generalized Abel integral equation. The applicability of the earlier proposed methods was restricted to the numerical inversion of a part of the generalized Abel integral equation. The method is quite accurate and stable as illustrated by applying it to intensity data with and without random noise to invert and compare it with the known analytical inverse. Thus it is a good method for applying to experimental intensities distorted by noise.

Operator Splitting Method for Coupled Problems:Transport and Maxwell Equations

Abstract: In this article a new approach is considered for implementing operator splitting methods for transport problems, influenced by electric fields. Our motivation came to model PE-CVD (plasma-enhanced chemical vapor deposition) processes, means the flow of species to a gas-phase, which are influenced by an electric field. Such a field we can model by wave equations. The main contributions are to improve the standard discretization schemes of each part of the coupling equation. So we discuss an improvement with implicit Runge- Kutta methods instead of the Yee’s algorithm. Further we balance the solver method between the Maxwell and Transport equation.

An Exponential Series Method for the Solution of Free Convection Boundary Layer Flow in a Saturated Porous Medium

Abstract: Third order nonlinear ordinary differential equation, subject to appropriate boundary conditions, arising in fluid mechanics is solved exactly using more suggestive schemes- Dirichlet series and method of stretching variables. These methods have advantages over pure numerical methods in obtaining derived quantities accurately for various values of the parameters involved at a stretch and are valid in a much larger domain compared with classical numerical schemes.

Context-Dependent Data Envelopment Analysis with Interval Data

Abstract: Data envelopment analysis (DEA) is a non-parametric method for evaluating the relative efficiency of decision making units (DMUs) on the basis of multiple inputs and outputs. The context-dependent DEA is introduced to measure the relative attractiveness of a particular DMU when compared to others. In real-world situation, because of incomplete or non-obtainable information, the data (Input and Output) are often not so deterministic, therefore they usually are imprecise data such as interval data, hence the DEA models becomes a nonlinear programming problem and is called imprecise DEA (IDEA). In this paper the context-dependent DEA models for DMUs with interval data is extended. First, we consider each DMU (which has interval data) as two DMUs (which have exact data) and then, by solving some DEA models, we can find intervals for attractiveness degree of those DMUs. Finally, some numerical experiment is used to illustrate the proposed approach at the end of paper.

Strong-Stability-Preserving, K-Step, 5- to 10-Stage, Hermite-Birkhoff Time-Discretizations of Order 12

Abstract: We construct optimal k-step, 5- to 10-stage, explicit, strong-stability-preserving Hermite-Birkhoff (SSP HB) methods of order 12 with nonnegative coefficients by combining linear k-step methods of order 9 with 5- to 10-stage Runge-Kutta (RK) methods of order 4. Since these methods maintain the monotonicity property, they are well suited for solving hyperbolic PDEs by the method of lines after a spatial discretization. It is seen that the 8-step 7-stage HB methods have largest effective SSP coefficient among the HB methods of order 12 on hand. On Burgers’ equations, some of the new HB methods have larger maximum effective CFL numbers than Huang’s 7-step hybrid method of order 7, thus allowing larger step size.

Conditional Value-at-Risk for Random Immediate Reward Variables in Markov Decision Processes

Abstract: We consider risk minimization problems for Markov decision processes. From a standpoint of making the risk of random reward variable at each time as small as possible, a risk measure is introduced using conditional value-at-risk for random immediate reward variables in Markov decision processes, under whose risk measure criteria the risk-optimal policies are characterized by the optimality equations for the discounted or average case. As an application, the inventory models are considered.

An Upwind Finite Volume Element Method for Nonlinear Convection Diffusion Problem

Abstract: A class of upwind finite volume element method based on tetrahedron partition is put forward for a nonlinear convection diffusion problem. Some techniques, such as calculus of variations, commutating operators and the a priori estimate, are adopted. The a priori error estimate in L2-norm and H1-norm is derived to determine the error between the approximate solution and the true solution.

A Gas Dynamics Method Based on the Spectral Deferred Corrections (SDC) Time Integration Technique and the Piecewise Parabolic Method (PPM)

Abstract: We present a computational gas dynamics method based on the Spectral Deferred Corrections (SDC) time integration technique and the Piecewise Parabolic Method (PPM) finite volume method. The PPM framework is used to define edge-averaged quantities, which are then used to evaluate numerical flux functions. The SDC technique is used to integrate solution in time. This kind of approach was first taken by Anita et al in [1]. However, [1] is problematic when it is implemented to certain shock problems. Here we propose significant improvements to [1]. The method is fourth order (both in space and time) for smooth flows, and provides highly resolved discontinuous solutions. We tested the method by solving variety of problems. Results indicate that the fourth order of accuracy in both space and time has been achieved when the flow is smooth. Results also demonstrate the shock capturing ability of the method.

Numerical Solution of Obstacle Problems by B-Spline Functions

Abstract: In this study, we use B-spline functions to solve the linear and nonlinear special systems of differential equations associated with the category of obstacle, unilateral, and contact problems. The problem can easily convert to an optimal control problem. Then a convergent approximate solution is constructed such that the exact boundary conditions are satisfied. The numerical examples and computational results illustrate and guarantee a higher accuracy for this technique.

A New Fourth Order Difference Approximation for the Solution of Three-Dimensional Non-Linear Biharmonic Equations Using Coupled Approach

Abstract: This paper deals with a new higher order compact difference scheme, which is, O(h4) using coupled approach on the 19-point 3D stencil for the solution of three dimensional nonlinear biharmonic equations. At each internal grid point, the solution u(x,y,z) and its Laplacian Δ4u are obtained. The resulting stencil algo-rithm is presented and hence this new algorithm can be easily incorporated to solve many problems. The present discretization allows us to use the Dirichlet boundary conditions only and there is no need to discretize the derivative boundary conditions near the boundary. We also show that special treatment is required to handle the boundary conditions. Convergence analysis for a model problem is briefly discussed. The method is tested on three problems and compares very favourably with the corresponding second order approximation which we also discuss using coupled approach.

Methods of Approximation in hpk Framework for ODEs in Time Resulting from Decoupling of Space and Time in IVPs

Abstract: The present study considers mathematical classification of the time differential operators and then applies methods of approximation in time such as Galerkin method ( GM ), Galerkin method with weak form ( / GM WF ), Petrov-Galerkin method ( PGM ), weighted residual method (WRY ), and least squares method or process ( LSM or LSP ) to construct finite element approximations in time. A correspondence is established between these integral forms and the elements of the calculus of variations: 1) to determine which methods of approximation yield unconditionally stable (variationally consistent integral forms, VC ) computational processes for which types of operators and, 2) to establish which integral forms do not yield unconditionally stable computations (variationally inconsistent integral forms, VIC ). It is shown that variationally consistent time integral forms in hpk framework yield computational processes for ODEs in time that are unconditionally stable, provide a mechanism of higher order global differentiability approximations as well as higher degree local approximations in time, provide control over approximation error when used as a time marching process and can indeed yield time accurate solutions of the evolution. Numerical studies are presented using standard model problems from the literature and the results are compared with Wilson’s  method as well as Newmark method to demonstrate highly meritorious features of the proposed methodology.

Bringing out Fluids Experiments from Laboratory to in Silico—A Journey of Hundred Years

Abstract: By making use of the developments in the fields of numerical methods, computational technology and fluid dynamics models, computational fluid dynamics (CFD) progress forward to play an active role today in various industrial, academic and research activities. In many cases, CFD simulations replace expensive and time consuming laboratory experiments successfully by allowing engineers and scientists to capture pressure, velocity and force distributions. Researchers are now able to test various theoretical conditions unavailable in the laboratory and CFD studies help them to get deeper insights on existing theories. The century-old history started just to solve some stress analysis problems numerically and today CFD methodology is being applied not only in fluid dynamics also in chemical engineering, mineral processing, fire engineering, sports, medical imaging and even in acoustics. This paper reviews the growth of CFD as a discipline and discusses its contemporary methodology.

Analytical Solution of Two Extended Model Equations for Shallow Water Waves by He's Variational Iteration Method

Abstract: In this paper, we consider two extended model equations for shallow water waves. We use He’s variational iteration method (VIM) to solve them. It is proved that this method is a very good tool for shallow water wave equations and the obtained solutions are shown graphically.