Some three-point boundary value problems for a second-order ordinary differential equation with variable coefficients are investigated in the present paper. By using the integration method, the second-order three-poin...
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Some three-point boundary value problems for a second-order ordinary differential equation with variable coefficients are investigated in the present paper. By using the integration method, the second-order three-point boundary value problems are transformed into a Fredholm integral equation of the second kind. The solutions and Green's functions for some special cases of the second-order three-point boundary value problems can be determined easily. The existence and uniqueness of the solutions of the given Fredholm integral equations are considered by using the fixed point theorem in Banach spaces. A new numerical method is further proposed to solve the second-kind Fredholm integral equation and an approximate solution is made. The convergence and error estimate of the obtained approximate solution are further analyzed. Numerical results are carried out to verify the feasibility and novelty of the proposed solution procedures. (C) 2014 Elsevier Inc. All rights reserved.
In this paper, by using the coincidence degree theory and the upper and lower solutions method, we deal with the existence of multiple solutions to three-point boundary value problems for second-order differential equ...
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In this paper, by using the coincidence degree theory and the upper and lower solutions method, we deal with the existence of multiple solutions to three-point boundary value problems for second-order differential equation with impulses at resonance. An example is given to show the validity of our results.
In this paper, we study three-point boundary value problems of the following fractional functional differential equations involving the Caputo fractional derivative: (C)D(alpha)u(t) = f(t, u(t), (C)D(beta)u(t)), 0 <...
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In this paper, we study three-point boundary value problems of the following fractional functional differential equations involving the Caputo fractional derivative: (C)D(alpha)u(t) = f(t, u(t), (C)D(beta)u(t)), 0 < t < 1, u'(0) = 0, u'(1) = lambda u'(eta), where D-C(alpha), D-C(beta) denote Caputo fractional derivatives, 2 < alpha < 3, 0 < beta < 1, eta is an element of (0, 1), 1 < lambda < 1/2 eta. We use the Green function to reformulate boundaryvalueproblems into an abstract operator equation. By means of the Schauder fixed point theorem and the Banach contraction principle, some existence results of solutions are obtained, respectively. As an application, some examples are presented to illustrate the main results.
The mathematical modeling of the decisive event of astrophysics, physiology, and many other areas of science and technology witness the involvement of singular boundaryvalueproblems. The nonlocal boundary conditions...
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The mathematical modeling of the decisive event of astrophysics, physiology, and many other areas of science and technology witness the involvement of singular boundaryvalueproblems. The nonlocal boundary conditions are more informative than local boundary conditions (initial conditions and two-pointboundary conditions) to evaluate some mathematical models. This article presents a collocation approach-based matrix technique to approximate the solution of the fusion of a class of singular differential equations subject to nonlocal three-pointboundary conditions. The proposed strategy utilizes the truncation of the series expansion of a function belonging to L2[0,1]$$ {L}_2\left[0,1\right] $$ in terms of Bernoulli polynomials. It transforms the singular boundaryvalueproblems into a set of nonlinear algebraic equations, which can be dealt with by any mathematical software. The Lipschitz condition on an equivalent completely continuous nonlinear operator has been used to prove the convergence analysis of the scheme. Some extremely nonlinear test examples are solved and provided in contrast with the exact solution. These numerical results are also examined against some existing numerical techniques to verify the applicability and significance of the proposed methodology. There are a few numerical examples that are application based but do not have exact solutions. In such cases, residual error norm is employed to measure the accuracy of the numerical strategies. The computational data demonstrate the superiority and validity of the proposed technique over existing numerical approaches.
This paper focuses on developing an efficient numerical approach based on Taylor-wavelets for solving three-point (nonlocal) singular boundaryvalueproblems. A special case of the considered problem, with strongly no...
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This paper focuses on developing an efficient numerical approach based on Taylor-wavelets for solving three-point (nonlocal) singular boundaryvalueproblems. A special case of the considered problem, with strongly nonlinear source term, arises in thermal explosion in a cylindrical reactor. The existence of a unique solution is thoroughly discussed for the considered problem. To establish the current method, an equivalent integral equation is constructed for the original problem to overcome the singularity at the origin. The evaluation of derivatives appearing in the model is also avoided in this way. Moreover, this scheme skips the integrals while reducing them into a system of nonlinear algebraic equations. Unlike other methods, this new approach does not require any linearization, discretization, perturbation, or evaluation of nonlinear terms separately. To the best of our knowledge, this is the first application of the wavelet-based method to the considered problem. The formulation of the proposed method is further supported by its convergence and error analysis. Some numerical examples are solved to validate the efficiency and robustness of the proposed method. Further, the computational convergence rate (COR) is reported for the first few examples to assist the obtained numerical solution. Moreover, the obtained numerical results are compared with those of existing techniques in the literature.
This paper investigates the following singular systems of nonlinear second-order three-point boundary value problems {-u ''' = f (t, v), t is an element of(0,1) -nu '' = g (t, u), t is an element o...
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ISBN:
(纸本)9783642279478;9783642279485
This paper investigates the following singular systems of nonlinear second-order three-point boundary value problems {-u ''' = f (t, v), t is an element of(0,1) -nu '' = g (t, u), t is an element of(0,1) u'(0) = nu'(0) = 0, U(1) = alpha u'(eta), nu(1) = alpha nu'(eta) Where eta is an element of (0,1), alpha < 0, Under some weaker conditions, the existence of positive solutions is obtained by applying the fixed point theorem of cone expansion and compression.
In this paper, an existence result of positive solutions for some three-point boundary value problems is obtained by using Leray-Schauder nonlinear alternative.
In this paper, an existence result of positive solutions for some three-point boundary value problems is obtained by using Leray-Schauder nonlinear alternative.
A nonlinear second-order ordinary differential equation with four cases of three-pointboundaryvalue conditions is studied by investigating the existence and approximation of solutions. First, the integration method ...
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A nonlinear second-order ordinary differential equation with four cases of three-pointboundaryvalue conditions is studied by investigating the existence and approximation of solutions. First, the integration method is proposed to transform the considered boundaryvalueproblems into Hammerstein integral equations. Second, the existence of solutions for the obtained Hammerstein integral equations is analyzed by using the Schauder fixed point theorem. The contraction mapping theorem in Banach spaces is further used to address the uniqueness of solutions. Third, the approximate solution of Hammerstein integral equations is constructed by using a new numerical method, and its convergence and error estimate are analyzed. Finally, some numerical examples are addressed to verify the given theorems and methods.
In this paper, the authors consider the following fractional high-order three-pointboundaryvalue problem: D(0+)(alpha)u(t) + f(t, u(t)) = 0, t is an element of (0, 1), u(0) = u'(0) = ... = u((n-2))(0) = 0, D(0+)...
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In this paper, the authors consider the following fractional high-order three-pointboundaryvalue problem: D(0+)(alpha)u(t) + f(t, u(t)) = 0, t is an element of (0, 1), u(0) = u'(0) = ... = u((n-2))(0) = 0, D(0+)(alpha-1)u(eta) = kD(0+)(alpha-1)u(1), where k > 1, eta is an element of (0, 1), n - 1 < alpha <= n, n >= 3, D-0+(alpha) is the standard Riemann-Liouville derivative of order alpha, and f : [0, 1] x [0,+infinity) -> [0,+infinity) is continuous. By using some fixed point index theorems on a cone for differentiable operators, the authors obtain the existence of positive solutions to the above boundaryvalue problem.
This paper discusses both the nonexistence of positive solutions for second-order three-point boundary value problems when the nonlinear term f（t, x, y） is superlinear in y at y = 0 and the existence of multiple pos...
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This paper discusses both the nonexistence of positive solutions for second-order three-point boundary value problems when the nonlinear term f（t, x, y） is superlinear in y at y = 0 and the existence of multiple positive solutions for second-order three-point boundary value problems when the nonlinear term f（t, x,y） is superlinear in x at ＋∞.
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