Abstract: Bringing forth a survey on recent higher order spline
techniques for solving boundary value problems in ordinary
differential equations. Here we have discussed the summary of the
articles since 2000 till date based on higher order splines like Septic,
Octic, Nonic, Tenth, Eleventh, Twelfth and Thirteenth Degree
splines. Comparisons of methods with own critical comments as
remarks have been included.
Abstract: We have considered the harmonic oscillations and general dynamic (pseudo oscillations) systems of theory generalized Green-Lindsay of couple-stress thermo-elasticity for isotropic, homogeneous elastic media. Approximate solution of some mixed boundary value problems for finite domain, bounded by the some closed surface are constructed.
Abstract: A novel method based on Genetic Algorithm to solve the boundary value problems (BVPs) of the Falkner–Skan equation over a semi-infinite interval has been presented. In our approach, we use the free boundary formulation to truncate the semi-infinite interval into a finite one. Then we use the shooting method based on Genetic Algorithm to transform the BVP into initial value problems (IVPs). Genetic Algorithm is used to calculate shooting angle. The initial value problems arisen during shooting are computed by Runge-Kutta Fehlberg method. The numerical solutions obtained by the present method are in agreement with those obtained by previous authors.
Abstract: In this paper, by constructing a special operator and using fixed point index theorem of cone, we get the sufficient conditions for symmetric positive solution of a class of nonlinear singular boundary value problems with p-Laplace operator, which improved and generalized the result of related paper.
Abstract: In this paper, by constructing a special set and utilizing fixed point index theory, we study the existence of solution for singular differential equation in Banach space, which improved and generalize the result of related paper.
Abstract: In this paper, by constructing a special non-empty closed convex set and utilizing M¨onch fixed point theory, we investigate the existence of solution for a class of fourth-order singular differential equation in Banach space, which improved and generalized the result of related paper.
Abstract: In this paper, by constructing a special operator and using fixed point index theorem of cone, we get the sufficient conditions for symmetric positive solution of a class of nonlinear singular boundary value problems with p-Laplace operator, which improved and generalized the result of related paper.
Abstract: In this paper, the existence, multiplicity and
noexistence of positive solutions for a class of semipositone
discrete boundary value problems with two parameters is
studied by applying nonsmooth critical point theory and
sub-super solutions method.
Abstract: In this paper a numerical algorithm is described for solving the boundary value problem associated with axisymmetric, inviscid, incompressible, rotational (and irrotational) flow in order to obtain duct wall shapes from prescribed wall velocity distributions. The governing equations are formulated in terms of the stream function ψ (x,y)and the function φ (x,y)as independent variables where for irrotational flow φ (x,y)can be recognized as the velocity potential function, for rotational flow φ (x,y)ceases being the velocity potential function but does remain orthogonal to the stream lines. A numerical method based on the finite difference scheme on a uniform mesh is employed. The technique described is capable of tackling the so-called inverse problem where the velocity wall distributions are prescribed from which the duct wall shape is calculated, as well as the direct problem where the velocity distribution on the duct walls are calculated from prescribed duct geometries. The two different cases as outlined in this paper are in fact boundary value problems with Neumann and Dirichlet boundary conditions respectively. Even though both approaches are discussed, only numerical results for the case of the Dirichlet boundary conditions are given. A downstream condition is prescribed such that cylindrical flow, that is flow which is independent of the axial coordinate, exists.
Abstract: Linear two-point boundary value problems of order
two are solved using cubic trigonometric B-spline interpolation
method (CTBIM). Cubic trigonometric B-spline is a piecewise
function consisting of trigonometric equations. This method is tested
on some problems and the results are compared with cubic B-spline
interpolation method (CBIM) from the literature. CTBIM is found to
approximate the solution slightly more accurately than CBIM if the
problems are trigonometric.
Abstract: In this paper, we develop quartic nonpolynomial
spline method for the numerical solution of third order two point
boundary value problems. It is shown that the new method gives
approximations, which are better than those produced by other spline
methods. Convergence analysis of the method is discussed through
standard procedures. Two numerical examples are given to illustrate
the applicability and efficiency of the novel method.
Abstract: In this work, we present a reliable framework to solve boundary value problems with particular significance in solid mechanics. These problems are used as mathematical models in deformation of beams. The algorithm rests mainly on a relatively new technique, the Variational Iteration Method. Some examples are given to confirm the efficiency and the accuracy of the method.
Abstract: Linear two-point boundary value problem of order two is solved using extended cubic B-spline interpolation method. There is one free parameters, λ, that control the tension of the solution curve. For some λ, this method produced better results than cubic B-spline interpolation method.
Abstract: In this paper, we study the existence of solution of
the four-point boundary value problem for second-order differential
equations with impulses by using Leray-Schauder theory:
Abstract: In this paper, a numerical solution based on sinc
functions is used for finding the solution of boundary value problems
which arise from the problems of calculus of variations. This
approximation reduce the problems to an explicit system of algebraic
equations. Some numerical examples are also given to illustrate the
accuracy and applicability of the presented method.
Abstract: A kind of singularly perturbed boundary value problems is under consideration. In order to obtain its approximation, simple upwind difference discretization is applied. We use a moving mesh iterative algorithm based on equi-distributing of the arc-length function of the current computed piecewise linear solution. First, a maximum norm a posteriori error estimate on an arbitrary mesh is derived using a different method from the one carried out by Chen [Advances in Computational Mathematics, 24(1-4) (2006), 197-212.]. Then, basing on the properties of discrete Green-s function and the presented posteriori error estimate, we theoretically prove that the discrete solutions computed by the algorithm are first-order uniformly convergent with respect to the perturbation parameter ε.
Abstract: Recently, the findings on the MEG iterative scheme has demonstrated to accelerate the convergence rate in solving any system of linear equations generated by using approximation equations of boundary value problems. Based on the same scheme, the aim of this paper is to investigate the capability of a family of four-point block iterative methods with a weighted parameter, ω such as the 4 Point-EGSOR, 4 Point-EDGSOR, and 4 Point-MEGSOR in solving two-dimensional elliptic partial differential equations by using the second-order finite difference approximation. In fact, the formulation and implementation of three four-point block iterative methods are also presented. Finally, the experimental results show that the Four Point MEGSOR iterative scheme is superior as compared with the existing four point block schemes.
Abstract: In this work, we solve multipoint boundary value
problems where the boundary value conditions are equations using
the Newton-Broyden Shooting method (NBSM).The proposed
method is tested upon several problems from the literature and the
results are compared with the available exact solution. The
experiments are given to illustrate the efficiency and implementation
of the method.
Abstract: In this paper , by using fixed point theorem , upper and lower solution-s method and monotone iterative technique , we prove the existence of maximum and minimum solutions of differential equations with delay , which improved and generalize the result of related paper.
Abstract: In this paper, a numerical solution based on nonpolynomial
cubic spline functions is used for finding the solution of
boundary value problems which arise from the problems of calculus
of variations. This approximation reduce the problems to an explicit
system of algebraic equations. Some numerical examples are also
given to illustrate the accuracy and applicability of the presented
method.