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Partial differential equation (PDE) based surfaces own a lot of advantages, compared to other types of 3D representation. For instance, fewer variables are required to represent the same 3D shape; the position, tangent, and even curvature continuity between PDE surface patches can be naturally maintained when certain conditions are satisfied, and the physics-based nature is also kept. Although some works applied implicit PDEs to 3D surface reconstruction from images, there is little work on exploiting the explicit solutions of PDE to this topic, which is more efficient and accurate. In this paper, we propose a new method to apply the explicit solutions of a fourth-order partial differential equation to surface reconstruction from multi-view images. The method includes two stages: point clouds data are extracted from multi-view images in the first stage, which is followed by PDE-based surface reconstruction from the obtained point clouds data. Our computational experiments show that the reconstructed PDE surfaces exhibit good quality and can recover the ground truth with high accuracy. A comparison between various solutions with different complexity to the fourth-order PDE is also made to demonstrate the power and flexibility of our proposed explicit PDE for surface reconstruction from images.

The objective of this paper is to study oscillation of fourth-order neutral differential equation. By using Riccati substitution and comparison technique, new oscillation conditions are obtained which insure that all solutions of the studied equation are oscillatory. Our results complement some known results for neutral differential equations. An illustrative example is included.

The purpose of this paper is to present several new results concerning relations between linear differential equations of the fourth order with one constraint and nonlinear ferential of the fourth order. We consider linear differential equations of the second, the third and the fourth order and nonlinear fourth order differential equations related via the Schwarzian derivative. The method is based on the use of the Schwarzian derivative, which is defined as the ratio of two linearly independent solutions of the linear differential equations of the second or third and fourth order. As a result we obtain new relations between the solutions of these linear and nonlinear equations. To illustrate theorems and our constructive approach we give two examples. The given method may be generalized to differential equations of higher orders.

This work aims to derive new inequalities that improve the asymptotic and oscillatory properties of solutions to fourth-order neutral differential equations. The relationships between the solution and its corresponding function play an important role in the oscillation theory of neutral differential equations. Therefore, we improve these relationships based on the modified monotonic properties of positive solutions. Additionally, we set new conditions that confirm the absence of positive solutions and thus confirm the oscillation of all solutions of the considered equation. We finally explain the importance of the new inequalities by applying our results to some special cases of the studied equation, as well as comparing them with previous results in the literature.

In this paper, we present a single-condition sharp criterion for the oscillation of the fourth-order linear delay differential equation x(4)(t)+p(t)x(τ(t))=0 by employing a novel method of iteratively improved monotonicity properties of nonoscillatory solutions. The result obtained improves a large number of existing ones in the literature.

In this paper, we consider a new distance structure, extended Branciari b-distance, to combine and unify several distance notions and obtain fixed point results that cover several existing ones in the corresponding literature. As an application of our obtained result, we present a solution for a fourth-order differential equation boundary value problem.

In the present paper, we consider the fourth-order differential equation u ( 4 ) ( x ) + ω u ′ ′ ( x ) + a ( x ) u ( x ) = f ( x , u ( x ) ) , ∀ x ∈ R ( 1 ) in which ω represents a constant, a ∈ C ( R , R ) and f ∈ C ( R 2 , R ) . We are concerned with the existence of ground state homoclinic solution for (1) when a is unnecessary positive and F ( x , u ) = ∫ 0 u f ( x , t ) d t satisfies a kind of superquadratic conditions due to Ding and Luan. For the proof, we apply a variant generalized weak linking theorem developed by Schechter and Zou. Some results in the literature are generalized and improved.

In this paper, new sufficient conditions for oscillation of fourth-order neutral differential equations are established. One objective of our paper is to further improve and complement some well-known results which were published recently in the literature. Symmetry ideas are often invisible in these studies, but they help us decide the right way to study them, and to show us the correct direction for future developments. An example is given to illustrate the importance of our results.

This article addresses the existence and uniqueness of solution for fully fourth-order differential equations modeling beams on elastic foundations with nonlinear boundary conditions. The proof will rely on Perov’s fixed point theorem in complete generalized metric spaces to overcome the problems due to the presence of all lower-order derivatives in the nonlinearity. Finally, some illustrating examples of the theory are presented.

The main purpose of this research was to use the comparison approach with a first-order equation to derive criteria for non-oscillatory solutions of fourth-order nonlinear neutral differential equations with p Laplacian operators. We obtained new results for the behavior of solutions to these equations, and we showed their symmetric and non-oscillatory characteristics. These results complement some previously published articles. To find out the effectiveness of these results and validate the proposed work, two examples were discussed at the end of the paper.