**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**10597

# Search results for: Calculus of variation; Non-polynomial spline functions; Numerical method

##### 10597 Non-Polynomial Spline Method for the Solution of Problems in Calculus of Variations

**Authors:**
M. Zarebnia,
M. Hoshyar,
M. Sedaghati

**Abstract:**

**Keywords:**
Calculus of variation; Non-polynomial spline functions; Numerical method

##### 10596 Quartic Nonpolynomial Spline Solutions for Third Order Two-Point Boundary Value Problem

**Authors:**
Talaat S. El-Danaf

**Abstract:**

**Keywords:**
Quartic nonpolynomial spline,
Two-point boundary
value problem.

##### 10595 A Review on Higher Order Spline Techniques for Solving Burgers Equation Using B-Spline Methods and Variation of B-Spline Techniques

**Authors:**
Maryam Khazaei Pool,
Lori Lewis

**Abstract:**

This is a summary of articles based on higher order B-splines methods and the variation of B-spline methods such as Quadratic B-spline Finite Elements Method, Exponential Cubic B-Spline Method Septic B-spline Technique, Quintic B-spline Galerkin Method, and B-spline Galerkin Method based on the Quadratic B-spline Galerkin method (QBGM) and Cubic B-spline Galerkin method (CBGM). In this paper we study the B-spline methods and variations of B-spline techniques to find a numerical solution to the Burgers’ equation. A set of fundamental definitions including Burgers equation, spline functions, and B-spline functions are provided. For each method, the main technique is discussed as well as the discretization and stability analysis. A summary of the numerical results is provided and the efficiency of each method presented is discussed. A general conclusion is provided where we look at a comparison between the computational results of all the presented schemes. We describe the effectiveness and advantages of these methods.

**Keywords:**
Burgers’ Equation,
Septic B-spline,
Modified Cubic
B-Spline Differential Quadrature Method,
Exponential Cubic
B-Spline Technique,
B-Spline Galerkin Method,
and Quintic B-Spline
Galerkin Method.

##### 10594 Sinc-Galerkin Method for the Solution of Problems in Calculus of Variations

**Authors:**
M. Zarebnia,
N. Aliniya

**Abstract:**

**Keywords:**
Calculus of variation; Sinc functions; Galerkin; Numerical method

##### 10593 Numerical Solution for Integro-Differential Equations by Using Quartic B-Spline Wavelet and Operational Matrices

**Authors:**
Khosrow Maleknejad,
Yaser Rostami

**Abstract:**

In this paper, Semi-orthogonal B-spline scaling functions and wavelets and their dual functions are presented to approximate the solutions of integro-differential equations.The B-spline scaling functions and wavelets, their properties and the operational matrices of derivative for this function are presented to reduce the solution of integro-differential equations to the solution of algebraic equations. Here we compute B-spline scaling functions of degree 4 and their dual, then we will show that by using them we have better approximation results for the solution of integro-differential equations in comparison with less degrees of scaling functions

**Keywords:**
Integro-differential equations,
Quartic B-spline
wavelet,
Operational matrices.

##### 10592 Septic B-Spline Collocation Method for Numerical Solution of the Kuramoto-Sivashinsky Equation

**Authors:**
M. Zarebnia,
R. Parvaz

**Abstract:**

In this paper the Kuramoto-Sivashinsky equation is solved numerically by collocation method. The solution is approximated as a linear combination of septic B-spline functions. Applying the Von-Neumann stability analysis technique, we show that the method is unconditionally stable. The method is applied on some test examples, and the numerical results have been compared with the exact solutions. The global relative error and L∞ in the solutions show the efficiency of the method computationally.

**Keywords:**
Kuramoto-Sivashinsky equation,
Septic B-spline,
Collocation
method,
Finite difference.

##### 10591 Unconventional Calculus Spreadsheet Functions

**Authors:**
Chahid K. Ghaddar

**Abstract:**

**Keywords:**
Calculus functions,
nonlinear systems,
differential algebraic equations,
solvers,
spreadsheet.

##### 10590 Spline Basis Neural Network Algorithm for Numerical Integration

**Authors:**
Lina Yan,
Jingjing Di,
Ke Wang

**Abstract:**

A new basis function neural network algorithm is proposed for numerical integration. The main idea is to construct neural network model based on spline basis functions, which is used to approximate the integrand by training neural network weights. The convergence theorem of the neural network algorithm, the theorem for numerical integration and one corollary are presented and proved. The numerical examples, compared with other methods, show that the algorithm is effective and has the characteristics such as high precision and the integrand not required known. Thus, the algorithm presented in this paper can be widely applied in many engineering fields.

**Keywords:**
Numerical integration,
Spline basis function,
Neural
network algorithm

##### 10589 Discontinuous Galerkin Method for Total Variation Minimization on Inpainting Problem

**Authors:**
Xijian Wang

**Abstract:**

**Keywords:**
finite element method,
discontinuous Galerkin method,
total variation minimization,
inpainting

##### 10588 Spline Collocation for Solving System of Fredholm and Volterra Integral Equations

**Authors:**
N. Ebrahimi,
J. Rashidinia

**Abstract:**

In this paper, numerical solution of system of Fredholm and Volterra integral equations by means of the Spline collocation method is considered. This approximation reduces the system of integral equations to an explicit system of algebraic equations. The solution is collocated by cubic B-spline and the integrand is approximated by the Newton-Cotes formula. The error analysis of proposed numerical method is studied theoretically. The results are compared with the results obtained by other methods to illustrate the accuracy and the implementation of our method.

**Keywords:**
Convergence analysis,
Cubic B-spline,
Newton-
Cotes formula,
System of Fredholm and Volterra integral equations.

##### 10587 Non-Polynomial Spline Solution of Fourth-Order Obstacle Boundary-Value Problems

**Authors:**
Jalil Rashidinia,
Reza Jalilian

**Abstract:**

**Keywords:**
Quintic non-polynomial spline,
Boundary formula,
Convergence,
Obstacle problems.

##### 10586 Quintic Spline Solution of Fourth-Order Parabolic Equations Arising in Beam Theory

**Authors:**
Reza Mohammadi,
Mahdieh Sahebi

**Abstract:**

**Keywords:**
Fourth-order parabolic equation,
variable coefficient,
polynomial quintic spline,
off-step points,
stability analysis.

##### 10585 Extended Cubic B-spline Interpolation Method Applied to Linear Two-Point Boundary Value Problems

**Authors:**
Nur Nadiah Abd Hamid,
Ahmad Abd. Majid,
Ahmad Izani Md. Ismail

**Abstract:**

Linear two-point boundary value problem of order two is solved using extended cubic B-spline interpolation method. There is one free parameters, λ, that control the tension of the solution curve. For some λ, this method produced better results than cubic B-spline interpolation method.

**Keywords:**
two-point boundary value problem,
B-spline,
extendedcubic B-spline.

##### 10584 Applying Element Free Galerkin Method on Beam and Plate

**Authors:**
Mahdad M’hamed,
Belaidi Idir

**Abstract:**

**Keywords:**
Numerical computation,
element-free Galerkin,
moving least squares,
meshless methods.

##### 10583 Cubic Trigonometric B-Spline Applied to Linear Two-Point Boundary Value Problems of Order Two

**Authors:**
Nur Nadiah Abd Hamid ,
Ahmad Abd. Majid,
Ahmad Izani Md. Ismail

**Abstract:**

**Keywords:**
trigonometric B-spline,
two-point boundary valueproblem,
spline interpolation,
cubic spline

##### 10582 Cantor Interpolating Spline to Design Electronic Mail Boxes

**Authors:**
Adil Al-Rammahi

**Abstract:**

Electronic mail is very important in present time. Many researchers work for designing, improving, securing, fasting, goodness and others fields in electronic mail. This paper introduced new algorithm to use Cantor sets and cubic spline interpolating function in the electronic mail design. Cantor sets used as the area (or domain) of the mail, while spline function used for designing formula. The roots of spline function versus Cantor sets used as the controller admin. The roots calculated by the numerical Newton – Raphson's method. The result of this algorithm was promised.

**Keywords:**
Cantor sets,
spline,
electronic mail design,
Newton – Raphson's method.

##### 10581 Evaluating Sinusoidal Functions by a Low Complexity Cubic Spline Interpolator with Error Optimization

**Authors:**
Abhijit Mitra,
Harpreet Singh Dhillon

**Abstract:**

We present a novel scheme to evaluate sinusoidal functions with low complexity and high precision using cubic spline interpolation. To this end, two different approaches are proposed to find the interpolating polynomial of sin(x) within the range [- π , π]. The first one deals with only a single data point while the other with two to keep the realization cost as low as possible. An approximation error optimization technique for cubic spline interpolation is introduced next and is shown to increase the interpolator accuracy without increasing complexity of the associated hardware. The architectures for the proposed approaches are also developed, which exhibit flexibility of implementation with low power requirement.

**Keywords:**
Arithmetic,
spline interpolator,
hardware design,
erroranalysis,
optimization methods.

##### 10580 A Comparison of Some Splines-Based Methods for the One-dimensional Heat Equation

**Authors:**
Joan Goh,
Ahmad Abd. Majid,
Ahmad Izani Md. Ismail

**Abstract:**

**Keywords:**
Heat equation,
Collocation based,
Cubic Bspline,
Extended cubic uniform B-spline.

##### 10579 Cubic Trigonometric B-spline Approach to Numerical Solution of Wave Equation

**Authors:**
Shazalina Mat Zin,
Ahmad Abd. Majid,
Ahmad Izani Md. Ismail,
Muhammad Abbas

**Abstract:**

The generalized wave equation models various problems in sciences and engineering. In this paper, a new three-time level implicit approach based on cubic trigonometric B-spline for the approximate solution of wave equation is developed. The usual finite difference approach is used to discretize the time derivative while cubic trigonometric B-spline is applied as an interpolating function in the space dimension. Von Neumann stability analysis is used to analyze the proposed method. Two problems are discussed to exhibit the feasibility and capability of the method. The absolute errors and maximum error are computed to assess the performance of the proposed method. The results were found to be in good agreement with known solutions and with existing schemes in literature.

**Keywords:**
Collocation method,
Cubic trigonometric B-spline,
Finite difference,
Wave equation.

##### 10578 A New Quadrature Rule Derived from Spline Interpolation with Error Analysis

**Authors:**
Hadi Taghvafard

**Abstract:**

**Keywords:**
Quadrature,
Spline interpolation,
Trapezoidal rule,
Numericalintegration,
Error analysis.

##### 10577 Numerical Solution of Riccati Differential Equations by Using Hybrid Functions and Tau Method

**Authors:**
Changqing Yang,
Jianhua Hou,
Beibo Qin

**Abstract:**

A numerical method for Riccati equation is presented in this work. The method is based on the replacement of unknown functions through a truncated series of hybrid of block-pulse functions and Chebyshev polynomials. The operational matrices of derivative and product of hybrid functions are presented. These matrices together with the tau method are then utilized to transform the differential equation into a system of algebraic equations. Corresponding numerical examples are presented to demonstrate the accuracy of the proposed method.

**Keywords:**
Hybrid functions,
Riccati differential equation,
Blockpulse,
Chebyshev polynomials,
Tau method,
operational matrix.

##### 10576 Application of a SubIval Numerical Solver for Fractional Circuits

**Authors:**
Marcin Sowa

**Abstract:**

**Keywords:**
Numerical method,
SubIval,
fractional calculus,
numerical solver,
circuit analysis.

##### 10575 Solving One-dimensional Hyperbolic Telegraph Equation Using Cubic B-spline Quasi-interpolation

**Authors:**
Marzieh Dosti,
Alireza Nazemi

**Abstract:**

In this paper, the telegraph equation is solved numerically by cubic B-spline quasi-interpolation .We obtain the numerical scheme, by using the derivative of the quasi-interpolation to approximate the spatial derivative of the dependent variable and a low order forward difference to approximate the temporal derivative of the dependent variable. The advantage of the resulting scheme is that the algorithm is very simple so it is very easy to implement. The results of numerical experiments are presented, and are compared with analytical solutions by calculating errors L2 and L∞ norms to confirm the good accuracy of the presented scheme.

**Keywords:**
Cubic B-spline,
quasi-interpolation,
collocation method,
second-order hyperbolic telegraph equation.

##### 10574 Septic B-spline Collocation Method for Solving One-dimensional Hyperbolic Telegraph Equation

**Authors:**
Marzieh Dosti,
Alireza Nazemi

**Abstract:**

Recently, it is found that telegraph equation is more suitable than ordinary diffusion equation in modelling reaction diffusion for such branches of sciences. In this paper, a numerical solution for the one-dimensional hyperbolic telegraph equation by using the collocation method using the septic splines is proposed. The scheme works in a similar fashion as finite difference methods. Test problems are used to validate our scheme by calculate L2-norm and L∞-norm. The accuracy of the presented method is demonstrated by two test problems. The numerical results are found to be in good agreement with the exact solutions.

**Keywords:**
B-spline,
collocation method,
second-order hyperbolic telegraph equation,
difference schemes.

##### 10573 A Comparison of the Nonparametric Regression Models using Smoothing Spline and Kernel Regression

**Authors:**
Dursun Aydin

**Abstract:**

**Keywords:**
Kernel regression,
Nonparametric models,
Prediction,
Smoothing spline.

##### 10572 The Boundary Element Method in Excel for Teaching Vector Calculus and Simulation

**Authors:**
Stephen Kirkup

**Abstract:**

**Keywords:**
Boundary element method,
laplace equation,
vector calculus,
simulation,
education.

##### 10571 Cubic B-spline Collocation Method for Numerical Solution of the Benjamin-Bona-Mahony-Burgers Equation

**Authors:**
M. Zarebnia,
R. Parvaz

**Abstract:**

In this paper, numerical solutions of the nonlinear Benjamin-Bona-Mahony-Burgers (BBMB) equation are obtained by a method based on collocation of cubic B-splines. Applying the Von-Neumann stability analysis, the proposed method is shown to be unconditionally stable. The method is applied on some test examples, and the numerical results have been compared with the exact solutions. The L∞ and L2 in the solutions show the efficiency of the method computationally.

**Keywords:**
Benjamin-Bona-Mahony-Burgers equation,
Cubic Bspline,
Collocation method,
Finite difference.

##### 10570 Numerical Algorithms for Solving a Type of Nonlinear Integro-Differential Equations

**Authors:**
Shishen Xie

**Abstract:**

In this article two algorithms, one based on variation iteration method and the other on Adomian's decomposition method, are developed to find the numerical solution of an initial value problem involving the non linear integro differantial equation where R is a nonlinear operator that contains partial derivatives with respect to x. Special cases of the integro-differential equation are solved using the algorithms. The numerical solutions are compared with analytical solutions. The results show that these two methods are efficient and accurate with only two or three iterations

**Keywords:**
variation iteration method,
decomposition method,
nonlinear integro-differential equations

##### 10569 Use of Technology to Improve Students’ Attitude in Learning Mathematics of Non-Mathematics Undergraduate Students

**Authors:**
Asia Majeed

**Abstract:**

This paper will investigate a form of learning mathematics by integrating technology in mathematics specifically for the university first-year calculus class to support students’ engagement in learning which influences students' conceptual and procedural understanding of the calculus content in a better way. The students with good grades in high school calculus generally struggle in first-year university calculus classes in learning mathematical analysis concepts. This problem has to be addressed. If this problem is not resolved, then most likely students with less ability to do mathematics might not able to complete their degrees. In this work, MATLAB is used to help students in learning and in improving calculus concepts.

**Keywords:**
Calculus,
first-year university students,
teaching strategies,
MATLAB.

##### 10568 Calculus-based Runtime Verification

**Authors:**
Xuan Qi,
Changzhi Zhao

**Abstract:**

**Keywords:**
calculus,
eagle logic,
monitor synthesis,
runtime verification