Existence of solutions for a third-order boundary value problem with p-Laplacian operator and nonlinear boundary conditions

Purpose The purpose of this paper is to solve the second-order nonlinear boundary value problem with nonlinear boundary conditions by an iterative numerical method. Design/methodology/approach The authors introduce eigenfunctions as test functions, such that a weak-form integral equation is derived. By expanding the numerical solution in terms of the weighted eigenfunctions and using the orthogonality of eigenfunctions with respect to a weight function, and together with the non-separated/mixed boundary conditions, one can obtain the closed-form expansion coefficients with the aid of Drazin inversion formula. Findings When the authors develop the iterative algorithm, removing the time-varying terms as well as the nonlinear terms to the right-hand sides, to solve the nonlinear boundary value problem, it is convergent very fast and also provides very accurate numerical solutions. Research limitations/implications Basically, the authors’ strategy for the iterative numerical algorithm is putting the time-varying terms as well as the nonlinear terms on the right-hand sides. Practical implications Starting from an initial guess with zero value, the authors used the closed-form formula to quickly generate the new solution, until the convergence is satisfied. Originality/value Through the tests by six numerical experiments, the authors have demonstrated that the proposed iterative algorithm is applicable to the highly complex nonlinear boundary value problems with nonlinear boundary conditions. Because the coefficient matrix is set up outside the iterative loop, and due to the property of closed-form expansion coefficients, the presented iterative algorithm is very time saving and converges very fast.

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Classical solution of a one-dimensional parabolic boundary value problem with nonlinear boundary conditions and moving boundary

Differential Equations ◽

10.1134/s0012266110070128 ◽

2010 ◽

Vol 46 (7) ◽

pp. 1053-1062 ◽

Cited By ~ 1

Author(s):

I. A. Chernov

Keyword(s):

Boundary Value Problem ◽

Boundary Conditions ◽

Classical Solution ◽

Nonlinear Boundary ◽

Moving Boundary ◽

Boundary Value ◽

Nonlinear Boundary Conditions ◽

Parabolic Boundary ◽

One Dimensional

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Positive Solutions for a Singular Elliptic Equation Arising in a Theory of Thermal Explosion

Mathematics ◽

10.3390/math9172173 ◽

2021 ◽

Vol 9 (17) ◽

pp. 2173

Author(s):

Song-Yue Yu ◽

Baoqiang Yan

Keyword(s):

Elliptic Equation ◽

Boundary Value Problem ◽

Boundary Conditions ◽

Thermal Explosion ◽

Comparison Principle ◽

Nonlinear Boundary ◽

Existence Of Solutions ◽

Nonlinear Boundary Conditions ◽

Uniqueness Of The Solution ◽

Thermal Explosion Model

In this paper, the thermal explosion model described by a nonlinear boundary value problem is studied. Firstly, we prove the comparison principle under nonlinear boundary conditions. Secondly, using the sub-super solution theorem, we prove the existence of a positive solution for the case K(x)>0, as well as the monotonicity of the maximal solution on parameter λ. Thirdly, the uniqueness of the solution for K(x)<0 is proved, as well as the monotonicity of the solutions on parameter λ. Finally, we obtain some new results for the existence of solutions, and the dependence on the λ for the case K(x) is sign-changing.

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Existence of solutions of third-order functional problems with nonlinear boundary conditions

The ANZIAM Journal ◽

10.1017/s1446181100013651 ◽

2004 ◽

Vol 46 (1) ◽

pp. 33-44 ◽

Cited By ~ 5

Author(s):

Alberto Cabada ◽

Seppo Heikkilä

Keyword(s):

Differential Equations ◽

Nonlinear Boundary ◽

Existence Of Solutions ◽

Functional Dependence ◽

Boundary Value ◽

Upper And Lower Solutions ◽

Initial Boundary ◽

Nonlinear Boundary Conditions ◽

Third Order ◽

Initial Boundary Value

AbstractIn this paper some existence results for third-order differential equations with nonlinear boundary value conditions are derived. Functional dependence in the data is allowed. In the proofs we use the method of upper and lower solutions, Schauder's fixed point theorem and results from Cabada and Heikkilä on third-order differential equations with linear and nonfunctional initial-boundary value conditions.

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