Singular Third-order Boundary Value Problem Research Articles

Numerical solution of singularly perturbed singular third order boundary value problems with nonclassical sinc method

In this paper, we employ a nonclassical sinc-collocation method to compute numerical solutions for singularly perturbed singular third-order boundary value problems prevalent in various scientific and engineering domains. Utilizing the sinc approximation offers a strategic advantage in navigating singularities, thus enabling an efficient computational strategy. Our method streamlines the solution process by converting singular boundary value problems into sets of linear equations, thereby improving computational efficiency. Moreover, its straightforward implementation adds to its robustness. We explore the convergence properties and error estimation of our proposed methods in detail. Finally, we provide two illustrative examples that demonstrate the effectiveness of our approach.

Numerical solution of third order singular boundary value problems with nonclassical SE-sinc-collocation and nonclassical DE-sinc-collocation

In this paper, we use the nonclassical SE-sinc-collocation and nonclassical DE-sinc-collocation methods for the numerical solution of singular third-order boundary value problems. The novelty of the approach is based on using the new non-classical weight function for sinc method instead of the classic ones. The convergence and error estimation of our methods is discussed. Several examples are solved and numerical results are compared with the existed methods, the obtained results demonstrate the validity of the obtained theoretical results and the efficiency of the our methods.

Singular $\phi $-Laplacian third-order BVPs with derivative dependance

This work is devoted to the existence of solutions for a class of singular third-order boundary value problem associated with a $\phi $-Laplacian operator and posed on the positive half-line; the nonlinearity also depends on the first derivative. The upper and lower solution method combined with the fixed point theory guarantee the existence of positive solutions when the nonlinearity is monotonic with respect to its arguments and may have a space singularity; however no Nagumo type condition is assumed. An example of application illustrates the applicability of the existence result.

UPPER AND LOWER SOLUTIONS FOR Φ-LAPLACIAN THIRD-ORDER BVPs ON THE HALF LINE

In this paper, we investigate the existence of positive solution for a class of singular third-order boundary value problem associated with a �-Laplacian operator and posed on the positive half-line: � (�(-x 00 )) 0 (t) + f(t,x(t)) = 0, t > 0, x(0) = µx 0 (0), x 0 (+1) = x 00 (+1) = 0 where µ � 0. By using the upper and lower solution approach and the fixed point theory, the existence of positive solutions is proved under a monotonic condition on f. The nonlinearity f may be singular at x = 0. An example of application is included to illustrate the main existence result.

Positive solutions to singular third-order boundary value problems on purely discrete time scales

over a finite discrete intervalft0; t1;:::; tn; tnC1; tnC2g. We prove the existence of a positive solution by means of the lower and upper solutions method and the Brouwer fixed point theorem in conjunction with perturbation methods to approximate regular problems.

Iterative Solutions of Singular Boundary Value Problems of Third-Order Differential Equation

By using the cone theory and the Banach contraction mapping principle, the existence and uniqueness results are established for singular third-order boundary value problems. The theorems obtained are very general and complement previous known results.

Positive solutions and eigenvalue intervals of a singular third-order boundary value problem

This paper studies positive solutions and eigenvalue intervals of a nonlinear third-order two-point boundary value problem. The nonlinear term is allowed to be singular with respect to both the time and space variables. By constructing a proper cone and a

Positive Solutions for Singular Third-Order Boundary Value Problem with Dependence on the First Order Derivative on the Half-Line

By using fixed-point index theory and a new fixed-point theorem in cones, some sufficient conditions for the existence of countably many positive solutions for singular third-order boundary value problem on the half-line are established, which are the complement of previously known results. The interesting point lies in the fact that the nonlinear term is allowed to depend on the first order derivative u′.

Existence and uniqueness of a nontrivial solution for a class of third-order nonlinear p -Laplacian m -point eigenvalue problems on time scales

In this paper, we study the existence and uniqueness of a nontrivial solution for the following third-order p -Laplacian m -point eigenvalue problems on time scales: { ( ϕ p ( u Δ ∇ ) ) ∇ + λ f ( t , u ( t ) , u Δ ( t ) ) = 0 , t ∈ ( 0 , T ) , α u ( 0 ) − β u Δ ( 0 ) = 0 , u ( T ) = ∑ i = 1 m − 2 a i u ( ξ i ) , u Δ ∇ ( 0 ) = 0 , where ϕ p ( s ) is p -Laplacian operator, i.e., ϕ p ( s ) = | s | p − 2 s , p > 1 , ϕ p − 1 = ϕ q , 1 p + 1 q = 1 , λ > 0 is a parameter, 0 < ξ 1 < ⋯ < ξ m − 2 < ρ ( T ) . We obtain several sufficient conditions of the existence and uniqueness of nontrivial solution of the above eigenvalue problems when λ is in some interval. Our approach is based on the Leray–Schauder nonlinear alternative. As an application, some examples to demonstrate our results are given. The conditions we used in the paper are different from those in [Z.C. Hao, L. Debnath, On eigenvalue intervals and eigenfunctions of fourth-order singular boundary value problems, Appl. Math. Lett. 18 (2005) 543–553; D. Jiang, R.P. Agarwal, A uniqueness and existence theorem for a singular third-order boundary value problem on [ 0 , + 1 ) , Appl. Math. Lett. 15 (2002) 445–451; S.H. Li, Positive solutions of nonlinear singular third-order two-point boundary value problem, J. Math. Anal. Appl. 323 (2006) 413–425; Z. Liu, J.S. Ume, D.R. Anderson, S.M. Kang, Twin monotone positive solutions to a singular nonlinear third-order differential equation, J. Math. Anal. Appl. 334 (2007) 299–313; Q. Yao, Positive solutions to a special singular second-order boundary value problem, Math. Comput. Model. (in press); Q. Yao, Successive iteration of positive solution for a discontinuous third-order boundary value problem, Comput. Math. Appl. 53 (2007) 741–749].

An existence theorem for a singular third-order boundary value problem on [0,+∞)

An existence result for a singular third-order boundary value problem is proved in this work. Here the nonlinearity is of the form f ( y ) = ( 1 − y ) λ g ( y ) , where λ > 0 and g ( y ) is continuous and positive on ( 0 , 1 ] , and the boundary conditions are y ( 0 ) = 0 , y ( + ∞ ) = 1 , y ′ ( + ∞ ) = y ″ ( + ∞ ) = 0 . The problem arises in the study of draining and coating flows.

A uniqueness and Existence theorem for a singular third-order boundary value Problem on [0, ∞)

It is proved that the singular third-order boundary value problem y‴ = f( y), y(0) = 0, y(+∞) = 1, y′(+∞) = y″(+∞) = 0, has a unique solution. Here f( y) = (1 − y) λ g( y), λ > 0, g( y) is positive and continuous on (0, 1]. The problem arises in the study of draining and coating flows.

Singular third-order boundary value problems

Singular third-order boundary value problems