scholarly journals Numerical Solutions of General Fourth Order Two Point Boundary Value Problems by Galerkin Method with Legendre Polynomials

2015 ◽  
Vol 62 (2) ◽  
pp. 103-108 ◽  
Author(s):  
Md Bellal Hossain ◽  
Md Shafiqul Islam
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Legendre Polynomials ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Boundary Value ◽  
Equivalent Form ◽  
Linear And Nonlinear

In this paper, Galerkin weighted residual method is presented to find the numerical solutions of the general fourth order linear and nonlinear differential equations with essential boundary conditions. For this, the given differential equations and the boundary conditions over arbitrary finite domain [a, b] are converted into its equivalent form over the interval [0, 1]. Here the Legendre polynomials, over the interval [0, 1], are chosen as trial functions satisfying the corresponding homogeneous form of the Dirichlet boundary conditions. Details matrix formulations are derived for solving linear and nonlinear boundary value problems (BVPs). Numerical examples for both linear and nonlinear BVPs are considered to verify the proposed formulation and the results obtained are compared. DOI: http://dx.doi.org/10.3329/dujs.v62i2.21973 Dhaka Univ. J. Sci. 62(2): 103-108, 2014 (July)

scholarly journals NUMERICAL SOLUTION OF A CLASS OF NONLINEAR SYSTEM OF SECOND-ORDER BOUNDARY-VALUE PROBLEMS: A FOURTH-ORDER CUBIC SPLINE APPROACH

2015 ◽  
Vol 20 (5) ◽  
pp. 681-700 ◽  
Author(s):  
Suheil A. Khuri ◽  
Ali M. Sayfy
Keyword(s):  
Numerical Solution ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Second Order ◽  
Spline Collocation ◽  
Extended System ◽  
Collocation Approach ◽  
Linear And Nonlinear

A cubic B-spline collocation approach is described and presented for the numerical solution of an extended system of linear and nonlinear second-order boundary-value problems. The system, whether regular or singularly perturbed, is tackled using a spline collocation approach constructed over uniform or non-uniform meshes. The rate of convergence is discussed theoretically and verified numerically to be of fourth-order. The efficiency and applicability of the technique are demonstrated by applying the scheme to a number of linear and nonlinear examples. The numerical solutions are contrasted with both analytical and other existing numerical solutions that exist in the literature. The numerical results demonstrate that this method is superior as it yields more accurate solutions.

scholarly journals On the Use of Piecewise Standard Polynomials in the Numerical Solutions of Fourth Order Boundary Value Problems

2014 ◽  
Vol 33 ◽  
pp. 53-64 ◽  
Author(s):  
Md. Shafiqul Islam ◽  
Md. Bellal Hossain
Keyword(s):  
Boundary Conditions ◽  
Exact Solutions ◽  
Fourth Order ◽  
Legendre Polynomials ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Approximate Solutions ◽  
Linear Combinations ◽  
Piecewise Continuous ◽  
The Given

This paper is devoted to find the numerical solutions of the fourth order linear and nonlinear differential equations using piecewise continuous and differentiable polynomials, such as Bernstein, Bernoulli and Legendre polynomials with specified boundary conditions. We derive rigorous matrix formulations for solving linear and non-linear fourth order BVP and special care is taken about how the polynomials satisfy the given boundary conditions. The linear combinations of each polynomial are exploited in the Galerkin weighted residual approximation. The derived formulation is illustrated through various numerical examples. Our approximate solutions are compared with the exact solutions, and also with the solutions of the existing methods. The approximate solutions converge to the exact solutions monotonically even with desired large significant digits. GANIT J. Bangladesh Math. Soc. Vol. 33 (2013) 53-64 DOI: http://dx.doi.org/10.3329/ganit.v33i0.17659

scholarly journals An iterative method for solving boundary value problems for second order differential equations

2016 ◽  
Vol 12 (8) ◽  
pp. 6489-6499
Author(s):  
Ymnah Salah Alruwaily
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Nonlinear Differential Equations ◽  
Boundary Value ◽  
Second Order ◽  
Robin Boundary Conditions ◽  
Second Order Differential Equations ◽  
Adomian Decomposition ◽  
Robin Boundary

The purpose of this paper is to investigate the application of the Adomian decomposition method (ADM) for solving boundary value problems for second-order differential equations with Robin boundary conditions. We first reformulate the boundary value problems for linear equations as a fixed point problems for a linear Fredholm integral operator, and then apply the ADM. We also extend our approach to include second-order nonlinear differential equations subject Robin boundary conditions.

Beam Vibrations With Time-Dependent Boundary Conditions

1950 ◽  
Vol 17 (4) ◽  
pp. 377-380
Author(s):  
R. D. Mindlin ◽  
L. E. Goodman
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Separation Of Variables ◽  
Boundary Value ◽  
Time Dependent ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Vibration Problems

Abstract A procedure is described for extending the method of separation of variables to the solution of beam-vibration problems with time-dependent boundary conditions. The procedure is applicable to a wide variety of time-dependent boundary-value problems in systems governed by linear partial differential equations.

scholarly journals Repeated alternating loading of a elastoplastic three-layer plate in a temperature field

2021 ◽  
Vol 21 (1) ◽  
pp. 60-75
Author(s):  
Eduard I. Starovoitov ◽  
◽  
Denis V. Leonenko ◽  
Keyword(s):  
Temperature Field ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Nonlinear Differential Equations ◽  
Boundary Value ◽  
Plastic Region ◽  
Local Load ◽  
Natural State ◽  
Elastic Plastic

Axisymmetric deformation of a three-layer circular plate under repeated alternating loading from the plastic region by a local load is considered. To describe kinematics of asymmetrical on the thickness of the plate pack is adopted the hypothesis of a broken line. In a thin elastic-plastic load-bearing layers are used the hypothesis of Kirchhoff. A non-linearly elastic relatively thick filler is incompressible in thickness. It is taken to be a hypothesis of Tymoshenko regarding the straightness and the incompressibility of the deformed normals with linear approximation of the displacements through the thickness layer. The work of the filler in the tangential direction is taken into account. The physical relations of stress-strain relations correspond to the theory of small elastic-plastic deformations. The effect of heat flow is taken into account. The temperature field in the plate was calculated by the formula obtained by averaging the thermophysical parameters over the thickness of the package. The system of differential equations of equilibrium under loading of the plate from the natural state is obtained by the Lagrange variational method. Boundary conditions on the plate contour are formulated. The solution of the corresponding boundary value problem is reduced to finding the three desired functions: deflection, shear and radial displacement of the shear surface of the filler. A non-uniform system of ordinary nonlinear differential equations is written for these functions. Its analytical iterative solution is obtained in Bessel functions by the method of elastic solutions of Ilyushin. In case of repeated alternating loading of the plate, the solution of the boundary value problem is constructed using the theory of variable loading of Moskvitin. In this case, the hypothesis of similarity of plasticity functions at each loading step is used. Their analytical form is taken independent of the point of unloading. However, the material constants included in the approximation formulas will be different. The cyclic hardening of the material of the bearing layers is taken into account. The parametric analysis of the obtained solutions under different boundary conditions in the case of a local load distributed in a circle is carried out. The influence of temperature and nonlinearity of layer materials on the displacements in the plate is numerically investigated.

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