Numerical Modelling of Application of Higher Order Accurate and Compact Numerical Scheme

The term ‗spline‘ is derived from the flexible device used by shipbuilder &draftsmen to draw a curve through pre-assigned points (knots) in such a way that not only the curve is continuous but also its slope and curvature are continuous functions .Draftsman attach the wooden or metal strip with weights called ducks, which can be adjusted to keep the strip in required shape. So weights are attached with the strip to keep it in the required shape. In order to resolve the problem of working with higher degree polynomials the idea of piecewise polynomial come into existence .Instead of using polynomial for the entire domain ,the function can be approximated by several polynomials defined over the sub-domains. A polynomial which is presented over a certain domain by means of several polynomials defined over its sub-domains called a piecewise polynomial. The piecewise polynomial approximation allows us to construct highly accurate approximations, but because some approximation functions are not smooth at the point connecting separate piecewise polynomial approximation. Sometimes, while the polynomial is continuous, it may not be continuously differentiable on the interval of approximation and the graph of the interpolant may not be smooth. Splines are an attempt to solve this problem. The underlying core of the Spline is its basis function. The defining feature of the basis function is k not sequence i x . Let X be a set of N+1 non decreasing real numbers. N N x ≤ x ≤ x ≤ ≤ x ≤ x 0 1 2 −1 ... .Here x s i ' are called knots , the set X is the knot sequence which represents the active area of real numbers line that defines the spline basis ,and the half –open interval[ ) 1 , i i+ x x the th i knot span. If the knots are equally spaced .,.( ) i 1 i ei x − x + is a constant for 0 ≤ i ≤ N − ),1 the knots vectors or the knot sequence is said to be uniform; otherwise, it is called non-uniform. Each spline function of degree k covers k +1 knots or k intervals. Spline methods are a high-performance alternative to solve partial differential equations (PDEs). This paper gives an overview on the principles of Spline methodology, shows their use and analyzes their performance in application examples, and discusses its merits. Tensors preserve the dimensional structure of a discretized PDE, which makes it possible to develop highly efficient computational solvers.

In this study, we consider the Spline method concerning the partial differential equations:

(1) (2)

The nonlinear part F(u) is of Spline type such that where Here A is a non-zero real number and is a positive number less than the space dimension n. The equations (4.1) and (2) can be rewritten in the form of the integral equations

(3) (4)

where and the associated unitary group U(t) is realized by the transform as where denotes the Fourier transform of g defined by The operators cos and sin are defined by replacing with and respectively. If the solution u of (1) or (3) has a decay at infinity and smoothness, it satisfies two conservation laws:

(5)

Where is the complex inner product in L2. Also the solution of (2) or (4) or satisfies the conservation law:

(6)

The main concern of this study is to establish the global well-posedness and scattering of radial solutions of the equations (1) and (2). The study of the global well-posedness (GWP) and scattering for the semi- relativistic equation (1) has not second authors of the present study showed GWP for ifand if n=1, for if , and small data scattering forifIn this study we tried to fill the gap for GWP under the assumption of radial symmetry. For further study like blowup of solutions, solitary waves, mean field limit problem for semi-relativistic equation, see the references. The first result is on the GWP for radial solutions of (3). Theorem 1. Let 1 for n = 3 and for . Let be radially symmetric and assume that is sufficiently small if. Then there exists a unique radial solutionsuch thatof (3) satisfying the energy and L2 conservations (5). We mean by and by . Hereafter, the space denotes for and its norm for some Banach space B. If , we use for with norm . We also denote for all by The next result is on the small data scattering of radial solutions of (4.3) for Theorem 2. Let for n = 3 and for . Then there is a real numberandsuch that

(7)

For fixed such and , let be radially symmetric data. Then if is sufficiently small, then there exists a unique radial solution to (4.4). Moreover, there exist radial functions and such that Where is the solutions to the Cauchy problem

(8)

In the definit ion of initial data space the space can be slightly weakened by the homogeneous Sobolev space . In fact, for. Let be the weak ened space. Then one can easily show that the solution and then the existence of scattering operator of (2) on a small neighborhood of the origin in. For details see Remark M below. (J?J) which follows from the relation between the weight and the L2 estimate of Bessel function such that For the finiteness, the assumption is inevitable because as and as . For more explicit formula, see the identity below. Hence for the present it seems hard to improve the range of for the small data scattering. From the perspective of negative result for the scattering1, it will be very interesting to show the scattering up to the value of greater than 1.

In the last few years another numerical technique has been increasingly used to solve mathematical models in engineering research, the spline Method. The spline Method has a few distinct advantages over the Finite Element and Finite Difference Methods. The advantage over the Finite Difference Method is that the spline Collocation Method provides a piecewise-continuous, closed form solution. An advantage over the Finite Element Method is that the spline collocation method procedure is simpler and easy to apply many problems involving differential equations. Our experimental results nicely confirm the excellent numerical approximation properties of Spline and their unique combination of high computational efficiency and low memory consumption, thereby showing huge improvements over standard finite-element methods.

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