A cubic Dulac function for a polynomial autonomous system on the plane with right-hand sides that are fourth-order nonlinearities

2000 ◽  
Vol 36 (4) ◽  
pp. 625-628
Author(s):  
A. A. Grin
Keyword(s):  
Fourth Order ◽  
Autonomous System ◽  
Dulac Function ◽  
Right Hand

On Solvability and Well-Posedness of Boundary Value Problems for Nonlinear Hyperbolic Equations of the Fourth Order

2008 ◽  
Vol 15 (3) ◽  
pp. 555-569
Author(s):  
Tariel Kiguradze
Keyword(s):  
Hyperbolic Equation ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Hyperbolic Equations ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Nonlinear Hyperbolic Equation ◽  
Well Posedness ◽  
Nonlinear Hyperbolic Equations ◽  
Right Hand

Abstract In the rectangle Ω = [0, a] × [0, b] the nonlinear hyperbolic equation 𝑢(2,2) = 𝑓(𝑥, 𝑦, 𝑢) with the continuous right-hand side 𝑓 : Ω × ℝ → ℝ is considered. Unimprovable in a sense sufficient conditions of solvability of Dirichlet, Dirichlet–Nicoletti and Nicoletti boundary value problems are established.

scholarly journals L-STABLE (4,2)-METHOD OF THE FOURTH ORDER FOR SOLVING STIFF PROBLEMS

2017 ◽  
Vol 17 (8) ◽  
pp. 59-68
Author(s):  
E.A. Novikov
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Maximal Order ◽  
Stiff Problems ◽  
System Of Differential Equations ◽  
Maximum Order ◽  
Order Of Accuracy ◽  
Right Hand ◽  
The Right

(M,k)-methods for solving stiff problems, in which on each step two times the right-hand side of the system of differential equations is calculated are investigated. It is shown that the maximum order of accuracy of the L-stable (m,2)-method is equal to four. (4,2)-method of maximal order is built.

scholarly journals New fixed point approach for a fully nonlinear fourth order boundary value problem

2018 ◽  
Vol 36 (4) ◽  
pp. 209-223 ◽  
Author(s):  
Dang Quang A ◽  
Ngo Thi Kim Quy
Keyword(s):  
Boundary Value Problem ◽  
Fourth Order ◽  
Linear Growth ◽  
Boundary Value ◽  
Iterative Solution ◽  
Fixed Point Approach ◽  
Right Hand ◽  
The Right ◽  
Uniqueness Of A Solution ◽  
Theoretical Results

In this paper we propose a method for investigating the solvability and iterative solution of a nonlinear fully fourth order boundary value problem. Namely, by the reduction of the problem to an operator equation for the right-hand side function we establish the existence and uniqueness of a solution and the convergence of an iterative process. Our method completely differs from the methods of other authors and does not require the condition of boundedness or linear growth of the right-hand side function on infinity. Many examples, where exact solutions of the problems are known or not, demonstrate the effectiveness of the obtained theoretical results.

Entropy solutions of Dirichlet problem for a class of degenerate anisotropic fourth-order equations with L1-right-hand sides

2002 ◽  
Vol 50 (5) ◽  
pp. 581-619 ◽  
Author(s):  
Alexander Kovalevsky ◽  
Francesco Nicolosi
Keyword(s):  
Dirichlet Problem ◽  
Fourth Order ◽  
Entropy Solutions ◽  
Right Hand ◽  
Fourth Order Equations

AN INVERSE PROBLEM FOR DETERMINING RIGHT HAND SIDE OF EQUATIONS FOR HYPERBOLIC EQUATION OF FOURTH ORDER

2019 ◽  
Vol 20 (2) ◽  
pp. 143-161
Author(s):  
Aysel T. Ramazanova ◽  
Hamlet F. Kuliyev ◽  
Arnd Roesch
Keyword(s):  
Inverse Problem ◽  
Hyperbolic Equation ◽  
Fourth Order ◽  
Right Hand

Continuity of weak solutions to nonlinear fourth-order equations with strengthened ellipticity via Wolff potentials

2019 ◽  
Vol 33 ◽  
pp. 33-49
Author(s):  
Mykhailo Voitovych
Keyword(s):  
Fourth Order ◽  
Elliptic Equations ◽  
Homogeneous Medium ◽  
Second Order ◽  
Generalized Solutions ◽  
Hölder Continuous ◽  
Right Hand ◽  
Fourth Order Equations ◽  
The Right ◽  
Second Order Elliptic Equations

In the present article nonlinear fourth-order equations in the divergence form with L^1-right-hand sides and the strengthened ellipticity condition on the coefficients are analyzed. Such equations, but with sufficiently regular right-hand sides, first appeared in the works of Professor I.V. Skrypnik concerning the regularity of generalized solutions for multidimensional nonlinear elliptic equations of high order. This class of equations correctly generalizes the corresponding nonlinear second-order elliptic equations with non-standard growth conditions on the coefficients, which are models for numerous physical phenomena in non-homogeneous medium. The main result of the article is a theorem on an estimation of oscillations in a ball of solutions to the given equations via the Wolff potentials of their right-hand sides. To prove this, we use the improved Kilpelainen-Maly method and pointwise potential estimates of functions related to special subclasses of Sobolev spaces, akin to the well-known De Giorgi classes. A new point is the verification that these classes contain superpositions of solutions and Moser logarithmic functions that include the Wolf potential of the right-hand side of the equation. As a corollary, a new result is obtained on the interior continuity of solutions to the equations with right-hand sides from the Kato class, which is characterized by the uniform convergence to zero of the corresponding Wolff potentials. Some important cases of fulfilling this condition are considered: the right-hand side of the equation belongs to the Morrey space with an index exceeding a certain limiting value, then the solutions are locally Holder continuous; if the right-hand side belongs to the borderline Lorentz-Zygmund classes, then the solutions are only locally continuous, but they are not Holder continuous in the domain. In the case when the summability exponents of the right-hand sides of the equations under consideration are less than the borderline values, there are examples of unbounded discontinuous solutions. These facts are exact analogues of the corresponding results in the theory of second-order elliptic equations.

scholarly journals Generalized Functions And Calculus Operators Of Mathematica Applied To Evaluation Of Influence Lines And Envelopes Of Statically Indeterminate Beams

2015 ◽  
Vol 15 (2) ◽  
Author(s):  
Ryszard Walentyński
Keyword(s):  
Fourth Order ◽  
Analytical Method ◽  
Analytical Form ◽  
Generalized Functions ◽  
Order Equation ◽  
Fourth Order Equation ◽  
Dirac Delta ◽  
Right Hand ◽  
Third Derivative

Abstract The paper presents an analytical method of finding functions of influence lines of statically indeterminate beams. There are presented solutions of a fourth order equation with a right hand side with second and third derivative of Dirac delta. There is shown that their solution are influence lines of moments and transverse forces. Moreover, thanks to Mathematica, analytical form of envelopes functions can be evaluated.

In the classroom: 4×4 magic squares

1962 ◽  
Vol 9 (7) ◽  
pp. 392-395
Author(s):  
Frances Hewitt
Keyword(s):  
Fourth Order ◽  
Good Health ◽  
Magic Squares ◽  
Magic Square ◽  
Right Hand

In his painting “Melencolia,” Albrecht Dürer used a fourth order magic square in the upper right hand corner. These 4×4 magic squares were supposed by Renaissance astrologers to combat melancholy. Magical squares are actually very unmagical. They are square arrays of numbers whose sums in all rows, columns, or diagonals are the same. They were called magic squares by the ancients because it was thought that they brought good health and luck.

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