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In this article, we studied the existence of solutions for a more general form of nonlinear fourth-order differential equations by using a new generalization of the Arzelá–Ascoli theorem and Schauder fixed theorem under easier and general conditions. Moreover, we provided some sufficient conditions on the nonlinear function that allowed us to deduce the nonexistence results. Finally, we outlined an example to illustrate our main results. [ABSTRACT FROM AUTHOR]

The application of block methods in the study of dynamical system is one area which needs more research. Therefore, this work presents block methods and its application to solve some problems in solid and fluid mechanics. The method was developed directly using the approaches of collocation and interpolation, and using power series polynomial as a trial solution. First, the system of linear equations is solved to obtain the unknown coefficients. The coefficients gotten are then substituted into the approximate solution to obtain continuous scheme. The continuous scheme, its first, second and third derivatives are evaluated at all the grid points to generate the block methods. The derived methods were applied to solve fourth-order boundary value problems of ordinary differential equations arising from beams and chemical problems. The results demonstrate the reliability and efficiency of the proposed method.

This paper contemplates a numerical investigation of the convection-diffusion type's fourth-order singularly perturbed linear and nonlinear boundary value problems. First, the considered linear fourth-order differential equation is converted into a strongly/weakly coupled singularly perturbed system (depending on the coefficient of the first-order derivative) of two ordinary differential equations with Dirichlet boundary conditions to solve the problem numerically. One of the equations is free from the perturbation parameter in the system. To obtain the solution for this system, we propose a numerical method of quadratic B -splines on an exponentially graded mesh. Convergence analysis shows that the proposed numerical scheme is second-order uniformly convergent in the discrete maximum norm. The nonlinear differential equation is linearized using the quasilinearization technique, and then the proposed approach is applied to the linearized problem. The theoretical outcomes are validated by executing the proposed method on three test problems. • It is challenging to obtain parameter-uniform results for higher-order singularly perturbed linear and nonlinear problems. • This paper presents a novel idea to resolve the boundary layers efficiently. • We present a numerical method to solve wild-behaving higher-order problems using quadratic spline basis functions. • An exponentially graded mesh is used to resolve the boundary layers. [ABSTRACT FROM AUTHOR]

This paper considers a numerical study for the fourth-order singularly perturbed boundary value problems. The associated differential equation is converted into a weakly coupled system of two singularly perturbed ordinary differential equations with Dirichlet boundary conditions to solve the problem numerically. In the system, one of the equations is independent of the perturbation parameter. To solve this system, we present a numerical technique of quadratic B-splines on an exponentially graded mesh. The established results show that the scheme is second-order uniformly convergent in the discrete maximum norm. The theoretical results are validated using the proposed method on two test problems. [ABSTRACT FROM AUTHOR]

Abstract In this article, we are interested in studying the asymptotic behavior of fourth-order neutral differential equations. Despite the growing interest in studying the oscillatory behavior of delay differential equations of second-order, fourth-order equations have received less attention. We get more than one criterion to check the oscillation by the generalized Riccati method and the integral average technique. Our results are an extension and complement to some results published in the literature. Examples are given to prove the significance of new theorems.

The purpose of this paper is to establish an existence theorem for a system of nonlinear fourth-order differential equations with two parameters $ \begin{eqnarray*} \left\{ \begin{array}{rcl} u^{(4)}+A(x)u& = &\lambda f(x, u, v, u'', v''), \ 0 0, \mu > 0 $ are two parameters and $ f, g: [0, 1]\times[0, \infty)\times[0, \infty)\times(-\infty, 0)\times(-\infty, 0) \rightarrow \mathbb{R} $ are two continuous functions satisfy the growth conditions.

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. [ABSTRACT FROM AUTHOR]

In this paper, we study the oscillation of solutions for a fourth-order neutral nonlinear differential equation, driven by a $p$-Laplace differential operator of the form \begin{equation*} \begin{cases} \left( r\left( t\right) \Phi _{p_{1}}[w^{\prime \prime \prime }\left( t\right) ]\right) ^{\prime }+q\left( t\right) \Phi _{p_{2}}\left( u\left( \vartheta \left( t\right) \right) \right) =0, & \\ r\left( t\right) >0,\ r^{\prime }\left( t\right) \geq 0,\ t\geq t_{0}>0, & \end{cases} \end{equation*} The oscillation criteria for these equations have been obtained. Furthermore, some examples are given to illustrate the criteria.

In this article, we are interested in studying the asymptotic behavior of fourth-order neutral differential equations. Despite the growing interest in studying the oscillatory behavior of delay differential equations of second-order, fourth-order equations have received less attention. We get more than one criterion to check the oscillation by the generalized Riccati method and the integral average technique. Our results are an extension and complement to some results published in the literature. Examples are given to prove the significance of new theorems. [ABSTRACT FROM AUTHOR]

By employing the Riccati substitution technique, we establish new oscillation criteria for a class of fourth-order neutral differential equations. Our new criteria complement a number of existing ones. An illustrative example is provided. [ABSTRACT FROM AUTHOR]

In this paper, we study the oscillation of solutions of fourth-order neutral delay differential equations in non-canonical form. By using Riccati transformation, we establish some new oscillation conditions. We provide some examples to examine the applicability of our results.

In this article, we prove some new oscillation theorems for fourth-order differential equations. New oscillation results are established that complement related contributions to the subject. We use the Riccati technique and the integral averaging technique to prove our results. As proof of the effectiveness of the new criteria, we offer more than one practical example.

The paper presents meshless methods based on the mixed Meshless Local Petrov- Galerkin approach used for solving linear fourth-order differential equations. In all the methods presented here, the primary variable and its derivatives up to the third order are approximated separately. Three different mixed meshless methods are derived by different choices of test and trial functions and are verified using available analytical and reference solutions. The numerical performance of the presented algorithms is demonstrated by several representative numerical examples. [ABSTRACT FROM AUTHOR]

Some new sufficient conditions are established for the oscillation of fourth order neutral differential equations with continuously distributed delay of the form r t N x ‴ t α ′ + ∫ a b q t , ϑ x β δ t , ϑ d ϑ = 0 , where t ≥ t 0 and N x t : = x t + p t x φ t . An example is provided to show the importance of these results.

The main purpose of this manuscript is to show asymptotic properties of a class of differential equations with variable coefficients r ν w ‴ ν β ′ + ∑ i = 1 j q i ν y κ g i ν = 0 , where ν ≥ ν 0 and w ν : = y ν + p ν y σ ν . By using integral averaging technique, we get conditions to ensure oscillation of solutions of this equation. The obtained results improve and generalize the earlier ones; finally an example is given to illustrate the criteria.

Some sufficient conditions are established for the oscillation of fourth order neutral differential equations of the form r t z ‴ t α ′ + q t x β σ t = 0 , where z t : = x t + p t x τ t . By using the technique of Riccati transformation and integral averaging method, we get conditions to ensure oscillation of solutions of this equation. 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. Moreover, the importance of the obtained conditions is illustrated via some examples.

In this paper, we deal with the asymptotics and oscillation of the solutions of fourth-order neutral differential equations of the form r t z ‴ t α ′ + q t x α g t = 0 , where z t : = x t + p t x δ t . By using a generalized Riccati transformation, we study asymptotic behavior and derive some new oscillation criteria. Our results extend and improve 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.

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.

In this paper, we study the singular fourth-order differential equation with a deviating argument: By using Mawhin's continuation theorem and some analytic techniques, we establish some criteria to guarantee the existence of positive periodic solutions. The significance of this paper is that g has a strong singularity at x = 0 and satisfies a small force condition at x = ∞, which is different from the known ones in the literature. [ABSTRACT FROM AUTHOR]

This paper deals with the oscillation of the fourth-order linear delay differential equation with a negative middle term under the assumption that all solutions of the auxiliary third-order differential equation are nonoscillatory. Examples are included to illustrate the importance of results obtained. [ABSTRACT FROM AUTHOR]

By means of the Schechter's Linking method, we study the existence of 2T-periodic solutions of the non-autonomous fourth-order ordinary differential equation $$ u''''-Au''-Bu-V_u(t,u)=0 $$ where $A>0$, $B>0$, $V(t,u)in mathbb{C}^1(mathbb{R}imesmathbb{R}, mathbb{R})$ is 2T-periodic in t and satisfies either $02$, or $u V_u(t,u)-2V(t,u)geq d_3|u|^r$ with $rgeq 1$.

Sufficient conditions for the boundedness and square integrability of solutions and their derivatives of certain fourth order nonlinear differential equation are given by means of the Lyapunov's second method. Our results obtained in this work, generalize existing results on fourth order nonlinear differential equations in the literature. For illustration, an example is also given. [ABSTRACT FROM AUTHOR]

In this paper, we study the gaps between zeros of nontrivial solutions of fourth order differential equations subject to some boundary conditions, and establish some sufficient conditions for (2, 2)-disconjugacy. The results, which cover some boundary value problems in bending of beams, will be proved by making use of a generalization of Hardy's inequality and some Opial and Wirtinger type inequalities. Some examples are considered to illustrate the main results. [ABSTRACT FROM AUTHOR]

In this paper, we mainly explore the phenomenon of concentration of homoclinic solutions for a class of nonperiodic fourth-order equations with sublinear indefinite nonlinearities. The proof is based on variational methods. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with the eigenvalue singular problem of the spectral type differential equations of the fourth-order self-adjoint differential operators where, . It has been reported in the literature that spectral type differential equations above have general solutions in terms of hypergeometric functions. In this work it is showed that the general solution in terms of hypergeometric functions reduces to Jacobi orthogonal polynomials (as eigenfunctions) in the case of eigenvalue singular problem. The corresponding eigenvalues are found. As application, the natural frequencies and mode shapes of mechanical transverse vibrations of a nonuniform singular structure are reported. [Copyright &y& Elsevier]

We develop a theory of 'special functions' associated with a certain fourth-order differential operator $\mathcal{D}_{\mu,\nu}$ on ℝ depending on two parameters μ, ν. For integers μ, ν≥−1 with μ+ ν∈2ℕ, this operator extends to a self-adjoint operator on L(ℝ, x d x) with discrete spectrum. We find a closed formula for the generating functions of the eigenfunctions, from which we derive basic properties of the eigenfunctions such as orthogonality, completeness, L-norms, integral representations, and various recurrence relations. This fourth-order differential operator $\mathcal{D}_{\mu,\nu}$ arises as the radial part of the Casimir action in the Schrödinger model of the minimal representation of the group O( p, q), and our 'special functions' give K-finite vectors. [ABSTRACT FROM AUTHOR]

This paper considers the existence of positive solutions of four-point boundary value problems for fourth-order ordinary differential equations with deviating arguments and p-Laplacian. We discuss such problems in the cases when the deviating arguments are delayed or advanced, what may concern optimization issues related to some technical problems. To obtain the existence results, a fixed point theorem for cones due to Avery and Peterson is applied. According to the Author's knowledge, the results are new. It is a first paper where a fixed point theorem for cones is applied to fourth-order differential equations with deviating arguments and p-Laplacian. An example is included to verify the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a Legendre Petrov–Galekin method for one-dimensional linear fourth-order differential equations. A Legendre Petrov–Galerkin and Chebyshev collocation method is developed for the nonlinear Kuramoto–Sivashinsky equation. Numerical results are presented to demonstrate the efficiency of the proposed schemes, and optimal rates of convergence in the -norm are rigorously derived. [ABSTRACT FROM AUTHOR]

In this paper He's variational iteration method is used to solve some examples of linear and non-linear forth-order boundary value problems. The first problem compared with homotopy analysis method solution and the other ones with the exact solution. The results show the high accuracy and speed of convergence of this method. It is found that the variational iteration method is a powerful method for solving of the non-linear equations. [ABSTRACT FROM AUTHOR]

This paper considers fourth-order differential equations with four-point boundary conditions and deviating arguments. We establish sufficient conditions under which such boundary-value problems have positive solutions. We discuss such problems in the cases when the deviating arguments are delayed or advanced. In order to obtain the existence of at least three positive solutions, we use a fixed-point theorem due to Avery and Peterson. To the authors’ knowledge, this is a first paper where the existence of positive solutions of boundary-value problems for fourth-order differential equations with deviating arguments is discussed. [ABSTRACT FROM AUTHOR]

Abstract: It is well known that for the discretization of the biharmonic operator with spectral methods (Galerkin, tau, or collocation) we have a condition number of , where N is the number of retained modes of approximations. This paper presents some efficient spectral algorithms, for reducing this condition number to , based on the Jacobi–Galerkin methods for fourth-order equations in one variable. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to systems with specially structured matrices that can be efficiently inverted. Jacobi–Galerkin methods for fourth-order equations in two dimension are considered. Numerical results indicate that the direct solvers presented in this paper are significantly more accurate at large N values than that based on the Chebyshev– and Legendre–Galerkin methods. [Copyright &y& Elsevier]

Abstract: Comment on the paper “A class of methods based on non-polynomial spline functions for the solution of a special fourth-order boundary-value problems with engineering applications” in Applied Mathematics and Computation 174 (2006) 1169–1180. The paper considered a class of two-point boundary-value problems of the form where f(x) and g(x) are continuous on [a, b], and A i and B i (i =1,2) are finite real constants. Here we correct some mistake in derivation of non-polynomial spline, boundary formulas, truncation errors, convergence analysis and computational experiments. [Copyright &y& Elsevier]

In this paper, we study the oscillation of solutions of fourth-order neutral delay differential equations in non-canonical form. By using Riccati transformation, we establish some new oscillation conditions. We provide some examples to examine the applicability of our results. [ABSTRACT FROM AUTHOR]

In this article, we prove some new oscillation theorems for fourth-order differential equations. New oscillation results are established that complement related contributions to the subject. We use the Riccati technique and the integral averaging technique to prove our results. As proof of the effectiveness of the new criteria, we offer more than one practical example. [ABSTRACT FROM AUTHOR]

Some new sufficient conditions are established for the oscillation of fourth order neutral differential equations with continuously distributed delay of the form r t N x ‴ t α ′ + ∫ a b q t , ϑ x β δ t , ϑ d ϑ = 0 , where t ≥ t 0 and N x t : = x t + p t x φ t . An example is provided to show the importance of these results. [ABSTRACT FROM AUTHOR]

The main purpose of this manuscript is to show asymptotic properties of a class of differential equations with variable coefficients r ν w ‴ ν β ′ + ∑ i = 1 j q i ν y κ g i ν = 0 , where ν ≥ ν 0 and w ν : = y ν + p ν y σ ν . By using integral averaging technique, we get conditions to ensure oscillation of solutions of this equation. The obtained results improve and generalize the earlier ones; finally an example is given to illustrate the criteria. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the asymptotics and oscillation of the solutions of fourth-order neutral differential equations of the form r t z ‴ t α ′ + q t x α g t = 0 , where z t : = x t + p t x δ t . By using a generalized Riccati transformation, we study asymptotic behavior and derive some new oscillation criteria. Our results extend and improve 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. [ABSTRACT FROM AUTHOR]

Some sufficient conditions are established for the oscillation of fourth order neutral differential equations of the form r t z ‴ t α ′ + q t x β σ t = 0 , where z t : = x t + p t x τ t . By using the technique of Riccati transformation and integral averaging method, we get conditions to ensure oscillation of solutions of this equation. 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. Moreover, the importance of the obtained conditions is illustrated via some examples. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]