• Journal of the Chungcheong Mathematical Society
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• v.25 no.3
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• pp.579-587
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• 2012

A fourth-order family of triparametric extensions of Jarratt's method are proposed in this paper to find a simple root of nonlinear algebraic equations. Convergence analysis including numerical experiments for various test functions apparently verifies the fourth-order convergence and asymptotic error constants.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.3 no.2
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• pp.51-67
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• 1999

The numerical solution of the Navier-Stokes equations in a constricted stepped channel problem has been obtained using a fourth order numerical method. Transformations are made to have a fine grid near the sharp corner and a long channel downstream. The derivatives in the Navier-Stokes equations are replaced by fourth order central differences which result a 29-point computational stencil. A procedure is used to avoid extra numerical boundary conditions near the solid walls. Results have been obtained for Reynolds numbers up to 1000.

• Journal of applied mathematics & informatics
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• v.28 no.3_4
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• pp.837-844
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• 2010

In this paper, a class of initial value problem of hyperbolic equations for fourth order with multiple characteristics is considered and can be solved analytically by variable transforms. Also, similar to Goursat's problem we present a direct integration technique for finding a new solutions of an inhomogeneous hyperbolic equation of fourth order such that the attached conditions are given on its multiple characteristics.

• Journal of the Korean Mathematical Society
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• v.56 no.5
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• pp.1419-1439
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• 2019

In this paper, we consider a class of noncooperative fourth-order elliptic systems involving nonlocal terms and critical growth in a bounded domain. With the help of Limit Index Theory due to Li [32] combined with the concentration compactness principle, we establish the existence of infinitely many solutions for the problem under the suitable conditions on the nonlinearity. Our results significantly complement and improve some recent results on the existence of solutions for fourth-order elliptic equations and Kirchhoff type problems with critical growth.

• Structural Engineering and Mechanics
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• v.17 no.6
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• pp.735-749
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• 2004

This work presents a direct integration scheme, based on a fourth order finite difference approach, for elastodynamics. The proposed scheme was chosen as an alternative for attenuating the errors due to the use of the central difference method, mainly when the time-step length approaches the critical time-step. In addition to eliminating the spurious numerical oscillations, the fourth order finite difference scheme keeps the advantages of the central difference method: reduced computer storage and no requirement of factorisation of the effective stiffness matrix in the step-by-step solution. A study concerning the stability of the fourth order finite difference scheme is presented. The Finite Element Method and the Boundary Element Method are employed to solve elastodynamic problems. In order to verify the accuracy of the proposed scheme, two examples are presented and discussed at the end of this work.

• Korean Journal of Mathematics
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• v.18 no.4
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• pp.489-496
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• 2010

In this paper we consider the Dirichlet problem for an fourth order elliptic equation on a open set in $R^N$. By using variational methods we obtain the multiplicity of nontrivial weak solutions for the fourth order elliptic equation.

• Journal of applied mathematics & informatics
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• v.4 no.1
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• pp.47-62
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• 1997

This paper is concerned with investigating the global as-ymptotic behavior of the zero solution of the initial-boundary value problem for a nonlinear fourth order wave equation, Moreover an esti-mate of the rate of decay of the solutions is obtained.

• Kyungpook Mathematical Journal
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• v.45 no.3
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• pp.349-356
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• 2005

In this paper, by using the Leray-Schauder continuation theorem and Wirtinger-type inequalities, we establish the existence and uniqueness theorems for two-point boundary value problems of a certain class of fourth-order nonlinear differential equations.

• Structural Engineering and Mechanics
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• v.89 no.2
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• pp.199-211
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• 2024

The loading capacity of engineering structures/components reduces after the initiation and propagation of crack eventually leads to the final failure. Hence, it becomes essential to deal with the crack and its effects at the design and simulation stages itself, by detecting the prone area of the fracture. The phase-field (PF) method has been accepted widely in simulating fracture problems in complex geometries. However, most of the PF methods are formulated with second order continuity theoryinvolving C⁰ continuity. In the present study, PF method based on fourth-order (i.e., higher order) theory, maintaining C¹ continuity has been proposed for ductile fracture simulation. The formulation includes fourth-order derivative terms of phase field variable, varying between 0 and 1. Applications of fourth-order PF theory to ductile fracture simulation resulted in novelty in this area. The proposed formulation is numerically solved using a two-dimensional finite element (FE) framework in 3-layered manner system. The solutions thus obtained from the proposed fourth order theory for different benchmark problems portray the improvement in the accuracy of the numerical results and are well matched with experimental results available in the literature. These results are also compared with second-order PF theory and a comparison study demonstrated the robustness of the proposed model in capturing ductile behaviour close to experimental observations.

• Journal of electromagnetic engineering and science
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• v.11 no.1
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• pp.11-15
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• 2011

Preserving high-order accuracy in high-order FDTD solutions across dielectric interfaces is very important for practical time-domain electromagnetic simulations. This paper presents a fourth-order accurate numerical boundary scheme for the planar dielectric interface to be used in the fourth-order FDTD method proposed earlier by the author. The interface scheme for the two-dimensional (2-D) transverse magnetic (TM) polarization case is derived and validated by monitoring the $L_2$ norm errors in the numerical solutions of a partially-filled cavity demonstrating its fourth-order convergence and long-time numerical stability in the presence of the planar dielectric interface.