Solving nonlinear third-order boundary value problems based-on boundary shape functions
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Chein-Shan Liu
Abstract
For a third-order nonlinear boundary value problem (BVP), we develop two novel methods to find the solutions, satisfying boundary conditions automatically. A boundary shape function (BSF) is created to automatically satisfy the boundary conditions, which is then employed to develop new numerical algorithms by adopting two different roles of the free function in the BSF. In the first type algorithm, we let the BSF be the solution of the BVP and the free function be a new variable. In doing so, the nonlinear BVP is certainly and exactly transformed to an initial value problem for the new variable with its terminal values as unknown parameters, whereas the initial conditions are given. In the second type algorithm, let the free functions be a set of complete basis functions and the corresponding boundary shape functions be the new bases. Since the solution already satisfies the boundary conditions automatically, we can apply a simple collocation technique inside the domain to determine the expansion coefficients and then the solution is obtained. For the general higher-order boundary conditions, the BSF method (BSFM) can easily and quickly find a very accurate solution. Resorting on the BSFM, the existence of solution is proved, under the Lipschitz condition for the ordinary differential equation system of the new variable. Numerical examples, including the singularly perturbed ones, confirm the high performance of the BSF-based numerical algorithms.
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Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
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Research funding: None declared.
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Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
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Articles in the same Issue
- Frontmatter
- Original Research Articles
- Coordinated target tracking in sensor networks by maximizing mutual information
- Legendre wavelet residual approach for moving boundary problem with variable thermal physical properties
- Computational study of intravenous magnetic drug targeting using implanted magnetizable stent
- Extended logistic map for encryption of digital images
- Valuation of the American put option as a free boundary problem through a high-order difference scheme
- Power minimization of gas transmission network in fully transient state using metaheuristic methods
- A priori error estimates for finite element approximations to transient convection-diffusion-reaction equations in fluidized beds
- Higher order rogue waves for the(3 + 1)-dimensional Jimbo–Miwa equation
- Dynamic characteristics of supersonic turbulent free jets from four types of circular nozzles
- New insights into singularity analysis
- Chaos and bifurcations in a discretized fractional model of quasi-periodic plasma perturbations
- Wavelet collocation methods for solving neutral delay differential equations
- Numerical solution of highly non-linear fractional order reaction advection diffusion equation using the cubic B-spline collocation method
- Solving nonlinear third-order boundary value problems based-on boundary shape functions
- Positive radial solutions for Dirichlet problems involving the mean curvature operator in Minkowski space
- Effect of outlet impeller diameter on performance prediction of centrifugal pump under single-phase and cavitation flow conditions
- A fractional-order ship power system: chaos and its dynamical properties
- Graphical structure of extended b-metric spaces: an application to the transverse oscillations of a homogeneous bar
- Hypergeometric fractional derivatives formula of shifted Chebyshev polynomials: tau algorithm for a type of fractional delay differential equations
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