Duality of variational problems with a new approach

2018 ◽  
Vol 52 (1) ◽  
pp. 79-93
Author(s):  
S.K. Padhan
Keyword(s):  
Research Work ◽  
Variational Problems ◽  
Second Order ◽  
Third Order ◽  
New Approach ◽  
Dual Problems ◽  
Duality Results ◽  
The Third ◽  
Second Order Duality

The present investigation introduces the third order duality in variational problems, as because, in certain situations, first and second order duality do not yield solutions but it succeeds in finding the desired results. The duality results for the pair of variational primal problems and their corresponding third order dual problems are demonstrated. Counterexamples are provided to justify the importance of the current research work. It is found that many reported results of the literature are particular cases of this paper.

Second order duality for variational problems involving generalized convexity

OPSEARCH ◽  
2014 ◽  
Vol 52 (3) ◽  
pp. 582-596 ◽  
Author(s):  
Anurag Jayswal ◽  
I. M. Stancu-Minasian ◽  
Sarita Choudhury
Keyword(s):  
Generalized Convexity ◽  
Variational Problems ◽  
Second Order ◽  
Second Order Duality

scholarly journals B-splines Algorithms for Solving Fredholm Linear Integro-Differential Equations

2004 ◽  
Vol 1 (2) ◽  
pp. 340-346
Author(s):  
Baghdad Science Journal
Keyword(s):  
Differential Equations ◽  
Spline Function ◽  
Cubic Spline ◽  
Second Order ◽  
Third Order ◽  
B Spline ◽  
Numerical Examples ◽  
B Splines ◽  
The Third ◽  
New Procedures

Algorithms using the second order of B -splines [B (x)] and the third order of B -splines [B,3(x)] are derived to solve 1' , 2nd and 3rd linear Fredholm integro-differential equations (F1DEs). These new procedures have all the useful properties of B -spline function and can be used comparatively greater computational ease and efficiency.The results of these algorithms are compared with the cubic spline function.Two numerical examples are given for conciliated the results of this method.

A Two-Dimensional Potential Flow Model of the Wave Field Generated by a Semisubmerged Body in Heaving Motion

1988 ◽  
Vol 32 (02) ◽  
pp. 83-91
Author(s):  
X. M. Wang ◽  
M. L. Spaulding
Keyword(s):  
Free Surface ◽  
Boundary Conditions ◽  
Potential Flow ◽  
Flow Model ◽  
Second Order ◽  
Two Dimensional ◽  
Third Order ◽  
The Third ◽  
Potential Flow Model ◽  
Good Agreement

A two-dimensional potential flow model is formulated to predict the wave field and forces generated by a sere!submerged body in forced heaving motion. The potential flow problem is solved on a boundary fitted coordinate system that deforms in response to the motion of the free surface and the heaving body. The full nonlinear kinematic and dynamic boundary conditions are used at the free surface. The governing equations and associated boundary conditions are solved by a second-order finite-difference technique based on the modified Euler method for the time domain and a successive overrelaxation (SOR) procedure for the spatial domain. A series of sensitivity studies of grid size and resolution, time step, free surface and body grid redistribution schemes, convergence criteria, and free surface body boundary condition specification was performed to investigate the computational characteristics of the model. The model was applied to predict the forces generated by the forced oscillation of a U-shaped cylinder. Numerical model predictions are generally in good agreement with the available second-order theories for the first-order pressure and force coefficients, but clearly show that the third-order terms are larger than the second-order terms when nonlinearity becomes important in the dimensionless frequency range 1≤ Fr≤ 2. The model results are in good agreement with the available experimental data and confirm the importance of the third order terms.

WENO-TVD schemes for hyperbolic conservation laws

Analysis ◽  
2007 ◽  
Vol 27 (1) ◽  
Author(s):  
Yousef Hashem Zahran
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Building Blocks ◽  
Second Order ◽  
Third Order ◽  
Weno Schemes ◽  
Hyperbolic Conservation ◽  
Systematic Assessment ◽  
The Third ◽  
Seventh Order

The purpose of this paper is twofold. Firstly we carry out a modification of the finite volume WENO (weighted essentially non-oscillatory) scheme of Titarev and Toro [14] and [15].This modification is done by using two fluxes as building blocks in spatially fifth order WENO schemes instead of the second order TVD flux proposed by Titarev and Toro [14] and [15]. These fluxes are the second order TVD flux [19] and the third order TVD flux [20].Secondly, we propose to use these fluxes as a building block in spatially seventh order WENO schemes. The numerical solution is advanced in time by the third order TVD Runge–Kutta method. A way to extend these schemes to general systems of nonlinear hyperbolic conservation laws, in one and two dimension is presented. Systematic assessment of the proposed schemes shows substantial gains in accuracy and better resolution of discontinuities, particularly for problems involving long time evolution containing both smooth and non-smooth features.

Contigious hyperquadrics of coequipped hyperbands sHm

2019 ◽  
pp. 126-132
Author(s):  
Yu. Popov
Keyword(s):  
Projective Space ◽  
Second Order ◽  
Third Order ◽  
Base Surface ◽  
The Third ◽  
Two Parameter ◽  
Order Contact

We consider hyperquadrics that are internally connected to coequipped hyperbands in the projective space. Specifically, a hyperquadric Qn1 tangent to a hyperplane at the point is called a contiguous hyper quadric of a hyperband if it has a second-order contact with the base surface of the hyperband. In a the third order differential neighborhood of the forming element of the hyperband, two two-parameter bundles of fields of adjoining hyperquadrics are internally invariantly joined, their equations are given in a dot frame. The set of hyperquadrics such that the plane and the plane of Cartan are conjugate with respect to hyperquadric Qn1 is considered. The condition is shown under which the normal of the 2nd kind and the Cartan plane are conjugate with respect to the hyperquadric Qn1 . In addition, the following theorem is proved: normalization of a coequipped regular hyperband has a semi-internal equipment if and only if its normals of the first and second kind are polarly conjugate with respect to the hyperquadric.

A further method for the evaluation of zeros of Bessel functions and some new asymptotic expansions for zeros of functions of large order

1951 ◽  
Vol 47 (4) ◽  
pp. 699-712 ◽  
Author(s):  
F. W. J. Olver
Keyword(s):  
Differential Equation ◽  
Bessel Functions ◽  
Numerical Technique ◽  
Third Order ◽  
New Approach ◽  
The Third ◽  
Zeros Of Bessel Functions ◽  
Zeros Of Functions ◽  
Linear Differential ◽  
Image Position

In a recent paper (1) I described a method for the numerical evaluation of zeros of the Bessel functions Jn(z) and Yn(z), which was independent of computed values of these functions. The essence of the method was to regard the zeros ρ of the cylinder functionas a function of t and to solve numerically the third-order non-linear differential equation satisfied by ρ(t). It has since been successfully used to compute ten-decimal values of jn, s, yn, s, the sth positive zeros* of Jn(z), Yn(z) respectively, in the ranges n = 10 (1) 20, s = 1(1) 20. During the course of this work it was realized that the least satisfactory feature of the new method was the time taken for the evaluation of the first three or four zeros in comparison with that required for the higher zeros; the direct numerical technique for integrating the differential equation satisfied by ρ(t) becomes unwieldy for the small zeros and a different technique (described in the same paper) must be employed. It was also apparent that no mere refinement of the existing methods would remove this defect and that a new approach was required if it was to be eliminated. The outcome has been the development of the method to which the first part (§§ 2–6) of this paper is devoted.

Perturbation treatment of the Hartree–Fock equations for the ground states of the boron-like ions

1969 ◽  
Vol 47 (7) ◽  
pp. 699-705 ◽  
Author(s):  
C. S. Sharma ◽  
R. G. Wilson
Keyword(s):  
Ground State ◽  
Atomic System ◽  
Ground States ◽  
Great Accuracy ◽  
Second Order ◽  
Expectation Values ◽  
Third Order ◽  
First Order ◽  
Hartree Fock ◽  
The Third

The first-order Hartree–Fock and unrestricted Hartree–Fock equations for the ground state of a five electron atomic system are solved exactly. The solutions are used to evaluate the corresponding second-order energies exactly and the third-order energies with great accuracy. The first-order terms in the expectation values of 1/r, r, r2, and δ(r) are also calculated.

scholarly journals Second-order L∞ variational problems and the ∞-polylaplacian

2020 ◽  
Vol 13 (2) ◽  
pp. 115-140 ◽  
Author(s):  
Nikos Katzourakis ◽  
Tristan Pryer
Keyword(s):  
Critical Point ◽  
Numerical Experiments ◽  
Lagrange Equation ◽  
Variational Problems ◽  
Second Order ◽  
Third Order ◽  
Fully Nonlinear ◽  
Euclidean Length ◽  
Special Cases

AbstractIn this paper we initiate the study of second-order variational problems in {L^{\infty}}, seeking to minimise the {L^{\infty}} norm of a function of the hessian. We also derive and study the respective PDE arising as the analogue of the Euler–Lagrange equation. Given {\mathrm{H}\in C^{1}(\mathbb{R}^{n\times n}_{s})}, for the functional\mathrm{E}_{\infty}(u,\mathcal{O})=\|\mathrm{H}(\mathrm{D}^{2}u)\|_{L^{\infty}% (\mathcal{O})},\quad u\in W^{2,\infty}(\Omega),\mathcal{O}\subseteq\Omega,{}the associated equation is the fully nonlinear third-order PDE\mathrm{A}^{2}_{\infty}u:=(\mathrm{H}_{X}(\mathrm{D}^{2}u))^{\otimes 3}:(% \mathrm{D}^{3}u)^{\otimes 2}=0.{}Special cases arise when {\mathrm{H}} is the Euclidean length of either the full hessian or of the Laplacian, leading to the {\infty}-polylaplacian and the {\infty}-bilaplacian respectively. We establish several results for (1) and (2), including existence of minimisers, of absolute minimisers and of “critical point” generalised solutions, proving also variational characterisations and uniqueness. We also construct explicit generalised solutions and perform numerical experiments.

Sign in / Sign up

Export Citation Format

Share Document