Multiquadric Quasi-interpolation Research Articles

Using adomian’s decomposition and multiquadric quasi-interpolation methods for solving Newell–Whitehead equation

Abstract In this paper, we study numerical solution of the Newell–Whitehead equation (NWE) by using Adomian’s method (ADM) and Multiquadric quasi-interpolation method. ADM has been extensively used to solve linear and nonlinear problems arising many interesting physical and engineering applications. NWE is derived to describe the envelope of modulated roll-solution in systems with two large extended or unbounded space directions. We will show the results with an example.

High accuracy multiquadric quasi-interpolation

In this paper, we propose a new multilevel univariate multiquadric (MQ) quasi-interpolation approach with higher approximation order compared with the existing MQ quasi-interpolations. The proposed approach includes two schemes, which are based on inverse multiquadric radial basis function (IMQ-RBF) interpolation, and Wu & Schaback’s MQ quasi-interpolation operator L D . Error analysis shows that our operators produce higher accuracy. Numerical examples demonstrate that the proposed MQ quasi-interpolation schemes are valid.

A kind of multiquadric quasi-interpolation operator satisfying any degree polynomial reproduction property to scattered data

In this paper, by virtue of using the linear combinations of the shifts of f ( x ) to approximate the derivatives of f ( x ) and Waldron’s superposition idea (2009), we modify a multiquadric quasi-interpolation with the property of linear reproducing to scattered data on one-dimensional space, such that a kind of quasi-interpolation operator L r + 1 f has the property of r + 1 ( r ∈ Z , r ≥ 0 ) degree polynomial reproducing and converges up to a rate of r + 2 . There is no demand for the derivatives of f in the proposed quasi-interpolation L r + 1 f , so it does not increase the orders of smoothness of f . Finally, some numerical experiments are shown to compare the approximation capacity of our quasi-interpolation operators with that of Wu–Schaback’s quasi-interpolation scheme and Feng–Li’s quasi-interpolation scheme.

Stability of multiquadric quasi-interpolation to approximate high order derivatives

Numerical simulation of the high order derivatives based on the sampling data is an important and basic problem in numerical approximation, especially for solving the differential equations numerically. The classical method is the divided difference method. However, it has been shown strongly unstable in practice. Actually, it can only be used to simulate the lower order derivatives in applications. To simulate the high order derivatives, this paper suggests a new method using multiquadric quasi-interpolation. The stability of the multiquadric quasi-interpolation method is compared with the classical divided difference method. Moreover, some numerical examples are presented to confirm the theoretical results. Both theoretical results and numerical examples show that the multiquadric quasi-interpolation method is much stabler than the divided difference method. This property shows that multiquadric quasi-interpolation method is an efficient tool to construct an approximation of high order derivatives based on scattered sampling data even with noise.

Identifying the temperature distribution in a parabolic equation with overspecified data using a multiquadric quasi-interpolation method

In this paper, we use a kind of univariate multiquadric quasi-interpolation to solve a parabolic equation with overspecified data, which has arisen in many physical phenomena. We obtain the numerical scheme by using the derivative of the quasi-interpolation to approximate the spatial derivative of the dependent variable and a simple forward difference to approximate the temporal derivative of the dependent variable. The advantage of the presented scheme is that the algorithm is very simple so it is very easy to implement. The results of the numerical experiment are presented and are compared with the exact solution to confirm the good accuracy of the presented scheme.

A kind of Bernoulli-type quasi-interpolation operator with univariate multiquadrics

In this paper, a kind of Bernoulli-type operator is proposed by combining a univariate multiquadric quasi-interpolation operator with the generalized Taylor polynomial. With an assumption on the shape-preserving parameter c, the convergence rate of the new operator is derived, which indicates that it could produce the desired precision. Numerical comparisons show that this method offers a higher degree of accuracy. Moreover, the associated algorithm is easily implemented.

A numerical method for one-dimensional nonlinear sine-Gordon equation using multiquadric quasi-interpolation

In this paper, we use a univariate multiquadric quasi-interpolation scheme to solve the one-dimensional nonlinear sine-Gordon equation that is related to many physical phenomena. We obtain a numerical scheme by using the derivative of the quasi-interpolation to approximate the spatial derivative and a difference scheme to approximate the temporal derivative. The advantage of the obtained scheme is that the algorithm is very simple so that it is very easy to implement. The results of numerical experiments are presented and compared with analytical solutions to confirm the good accuracy of the presented scheme.

A kind of improved univariate multiquadric quasi-interpolation operators

In this paper, a kind of improved univariate multiquadric quasi-interpolation operators is proposed by using Hermite interpolating polynomials. Error analysis shows that the convergence rate of the operators depends heavily on the shape parameter c , which indicates that our operators could provide the desired smoothness and precision by choosing a suitable value of c . Numerical examples show that the operators provide a high degree of accuracy. Moreover, operators are applied to the fitting of discrete solutions of initial value problems.

A multiquadric quasi-interpolation with linear reproducing and preserving monotonicity

In this paper, we develop a multiquadric (MQ) quasi-interpolation which has the properties of linear reproducing and preserving monotonicity. Moreover, we give its approximation error by theoretic analysis and illustrate the effect by means of two examples. One of the examples is to approach the linear combination of two sine functions with different frequencies. Another is to approximate a function with discontinuity. From the results of the examples, we believe that the present MQ quasi-interpolation is feasible.

Solving partial differential equation by using multiquadric quasi-interpolation

In this paper, we use a kind of univariate multiquadric (MQ) quasi-interpolation to solve partial differential equation (PDE). 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. Our numerical experiment includes two examples. One is solving viscid Burgers’ equation for initial trapezoidal conditions. Another is simulating the interaction of two waves travelling in opposite direction. From the numerical experiment, we can see that the present scheme is valid.

Solving hyperbolic conservation laws using multiquadric quasi-interpolation

In this article, we apply the univariate multiquadric (MQ) quasi-interpolation to solve the hyperbolic conservation laws. At first we construct the MQ quasi-interpolation corresponding to periodic and inflow-outflow boundary conditions respectively. Next we obtain the numerical schemes to solve the partial differential equations, by using the derivative of the quasi-interpolation to approximate the spatial derivative of the differential equation and a low-order explicit difference to approximate the temporal derivative of the differential equation. Then we verify our scheme for the one-dimensional Burgers' equation (without viscosity). We can see that the numerical results are very close to the exact solution and the computational accuracy of the scheme is 𝒪(τ), where τ is the temporal step. We can improve the accuracy by using the high-order quasi-interpolation. Moreover the methods can be generalized to the other equations. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006

Applying multiquadric quasi-interpolation to solve Burgers’ equation

Abstract In this paper, we develop a kind of univariate multiquadric (MQ) quasi-interpolation and use it to solve Burgers’ equation (with viscosity). At first we construct the MQ quasi-interpolation, which possesses the properties of linear reproducing and preserving monotonicity. Next 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 time derivative of the dependent variable. Then, we verify our method for two examples with distinguishing initial value condition. One example is tested for three Reynolds number, that is, R = 10, R = 100, and R = 10,000. From the numerical experiments, we see that the presented method in this paper is valid although the accuracy of the technique is not higher than Hon and Mao’s one. Another example is used to examine the travelling of the shock. We can improve the accuracy by selecting an appropriate shape parameter and using a higher accurate MQ quasi-interpolation. The advantage of the resulting scheme is that the algorithm is very simple, so it is very easy to implement.

Shape preserving properties and convergence of univariate multiquadric quasi-interpolation

Shape preserving properties and convergence of univariate multiquadric quasi-interpolation