scholarly journals Research of derivatives of linear combination of fourth order B-splines

2016 ◽  
Vol 20 (0) ◽  
Author(s):  
P. O. Prystavka ◽  
O. H. Cholyshkina ◽  
B. I. Martuk
Keyword(s):  
Linear Combination ◽  
Fourth Order ◽  
B Splines ◽  
Derivatives Of

scholarly journals On a Set of Conform-Invariant Equations of the Gravitational Field

1953 ◽  
Vol 10 (1) ◽  
pp. 16-20 ◽  
Author(s):  
H. A. Buchdahl
Keyword(s):  
Gravitational Field ◽  
Linear Combination ◽  
Curvature Tensor ◽  
Wide Class ◽  
Ricci Tensor ◽  
Empty Space ◽  
Image Position ◽  
Derivatives Of

Eddington has considered equations of the gravitational field in empty space which are of the fourth differential order, viz. the sets of equations which express the vanishing of the Hamiltonian derivatives of certain fundamental invariants. The author has shown that a wide class of such equations are satisfied by any solution of the equationswhere Gμν and gμν are the components of the Ricci tensor and the metrical tensor respectively, whilst λ is an arbitrary constant. For a V4 this applies in particular when the invariant referred to above is chosen from the setwhere Bμνσρ is the covariant curvature tensor. K3 has been included since, according to a result due to Lanczos3, its Hamiltonian derivative is a linear combination of and , i.e. of the Hamiltonian derivatives of K1 and K2. In fact

scholarly journals Hexagonal Grid Computation of the Derivatives of the Solution to the Heat Equation by Using Fourth-Order Accurate Two-Stage Implicit Methods

2021 ◽  
Vol 5 (4) ◽  
pp. 203
Author(s):  
Suzan Cival Buranay ◽  
Nouman Arshad ◽  
Ahmed Hersi Matan
Keyword(s):  
Heat Equation ◽  
Fourth Order ◽  
Boundary Value ◽  
Time Variable ◽  
Implicit Methods ◽  
First Order ◽  
Mixed Derivatives ◽  
Hexagonal Grids ◽  
Spatial Derivatives ◽  
Derivatives Of

We give fourth-order accurate implicit methods for the computation of the first-order spatial derivatives and second-order mixed derivatives involving the time derivative of the solution of first type boundary value problem of two dimensional heat equation. The methods are constructed based on two stages: At the first stage of the methods, the solution and its derivative with respect to time variable are approximated by using the implicit scheme in Buranay and Arshad in 2020. Therefore, Oh4+τ of convergence on constructed hexagonal grids is obtained that the step sizes in the space variables x1, x2 and in time variable are indicated by h, 32h and τ, respectively. Special difference boundary value problems on hexagonal grids are constructed at the second stages to approximate the first order spatial derivatives and the second order mixed derivatives of the solution. Further, Oh4+τ order of uniform convergence of these schemes are shown for r=ωτh2≥116,ω>0. Additionally, the methods are applied on two sample problems.

scholarly journals The Solution of Nonlinear Fourth-Order Differential Equation with Integral Boundary Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Yanli Fu ◽  
Huanmin Yao
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Fourth Order ◽  
Reproducing Kernel ◽  
Order Differential Equation ◽  
Integral Boundary Conditions ◽  
Reproducing Kernel Space ◽  
Integral Boundary ◽  
Derivatives Of ◽  
Fourth Order Differential Equation

An iterative algorithm is proposed for solving the solution of a nonlinear fourth-order differential equation with integral boundary conditions. Its approximate solutionun(x)is represented in the reproducing kernel space. It is proved thatun(x)converges uniformly to the exact solutionu(x). Moreover, the derivatives ofun(x)are also convergent to the derivatives ofu(x). Numerical results show that the method employed in the paper is valid.

Approximation of 3D convection diffusion equation using DQM based on modified cubic trigonometric B-splines

2020 ◽  
pp. 1-10
Author(s):  
Mohammad Tamsir ◽  
Neeraj Dhiman ◽  
F.S. Gill ◽  
Robin
Keyword(s):  
Diffusion Equation ◽  
Numerical Solutions ◽  
Basis Functions ◽  
B Spline ◽  
Convection Diffusion ◽  
B Splines ◽  
Convection Diffusion Equation ◽  
First Order ◽  
The Stability ◽  
Derivatives Of

This paper presents an approximate solution of 3D convection diffusion equation (CDE) using DQM based on modified cubic trigonometric B-spline (CTB) basis functions. The DQM based on CTB basis functions are used to integrate the derivatives of space variables which transformed the CDE into the system of first order ODEs. The resultant system of ODEs is solved using SSPRK (5,4) method. The solutions are approximated numerically and also presented graphically. The accuracy and efficiency of the method is validated by comparing the solutions with existing numerical solutions. The stability analysis of the method is also carried out.

scholarly journals Volume derivatives of the Grüneisen parameter in the limit of infinite pressure

2019 ◽  
Vol 97 (1) ◽  
pp. 114-116 ◽  
Author(s):  
A. Dwivedi
Keyword(s):  
Boundary Condition ◽  
Fourth Order ◽  
Fundamental Theorem ◽  
High Pressures ◽  
Grüneisen Parameter ◽  
Third Order ◽  
Gruneisen Parameter ◽  
The Third ◽  
Properties Of Materials ◽  
Derivatives Of

Expressions have been obtained for the volume derivatives of the Grüneisen parameter, which is directly related to the thermal and elastic properties of materials at high temperatures and high pressures. The higher order Grüneisen parameters are expressed in terms of the volume derivatives, and evaluated in the limit of infinite pressure. The results, that at extreme compression the third-order Grüneisen parameter remains finite and the fourth-order Grüneisen parameter tends to zero, have been used to derive a fundamental theorem according to which the volume derivatives of the Grüneisen parameter of different orders, all become zero in the limit of infinite pressure. However, the ratios of these derivatives remain finite at extreme compression. The formula due to Al’tshuler and used by Dorogokupets and Oganov for interpolating the Grüneisen parameter at intermediate compressions has been found to satisfy the boundary condition at infinite pressure obtained in the present study.

scholarly journals RELATIONS BETWEEN HYPERELLIPTIC INTEGRALS

1994 ◽  
Vol 09 (19) ◽  
pp. 1791-1797 ◽  
Author(s):  
S. PAKULIAK ◽  
A. PERELOMOV
Keyword(s):  
Linear Combination ◽  
Simple Property ◽  
Hyperelliptic Surfaces ◽  
Level Zero ◽  
Branching Points ◽  
Arbitrary Genus ◽  
Derivatives Of ◽  
Zamolodchikov Equation

A simple property of the integrals over the hyperelliptic surfaces of arbitrary genus is observed. Namely, the derivatives of these integrals with respect to the branching points are given by the linear combination of the same integrals. We check that this property is responsible for the solution to the level zero Knizhnik-Zamolodchikov equation given in terms of hyperelliptic integrals.

NUMERICAL SOLUTION FOR THE NONLINEAR EMDEN–FOWLER TYPE EQUATIONS BY A FOURTH-ORDER ADAPTIVE METHOD

2013 ◽  
Vol 11 (01) ◽  
pp. 1350052 ◽  
Author(s):  
S. A. KHURI ◽  
A. SAYFY
Keyword(s):  
Finite Element ◽  
Numerical Solution ◽  
Rate Of Convergence ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Adaptive Method ◽  
B Splines ◽  
Special Cases ◽  
Collocation Approach

A finite element collocation approach, based on cubic B-splines, is manipulated for obtaining numerical solutions of a generalized form of the Emden–Fowler type equations. The rate of convergence is discussed theoretically and verified numerically to be of fourth-order by using the double-mesh principle. The efficiency of the scheme is tested on a number of examples which represent special cases of the problem under consideration. The results are compared with analytical and other numerical solutions that are available in the literature. The proposed method reveals that the outcomes are reliable and very accurate when contrasted with other existing methods.

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