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In this work, a new alternative uniformly convergent iterative scheme is presented and applied for the solution of an extended class of linear and nonlinear third order boundary value problems that arise in physical applications. The method is based on embedding Green’s functions into well-known fixed point iterations, including Picard’s and Krasnoselskii–Mann’s schemes. Convergence of the numerical method is proved by manipulating the contraction principle. The effectiveness of the proposed approach is established by implementing it on several numerical examples, including linear and nonlinear third order boundary value problems. The results show highly accurate approximations when compared to exact and existing numerical solutions. [ABSTRACT FROM AUTHOR]

Abstract: A numerical method based on finite differences and spline collocation is proposed for solving a generalized parabolic problem subject to non-classical conditions. It is shown numerically that this scheme has second-order convergence. The stability conditions relating the time step size to mesh sizes were examined through numerical examples. It turns out that the stability of the scheme in certain cases is subjected to a stability condition and depends on the nature of the problem. A number of specific test problems are solved to assess the efficiency and performance of the method. The numerical results obtained indicate that the approach is viable and yields accurate solutions when compared with the existing closed-form solutions and generates results consistent with recent numerical approaches. [Copyright &y& Elsevier]

Abstract: In the present paper, a collocation finite element approach based on cubic splines is presented for the numerical solution of a generalized wave equation subject to non-local conservation condition. The efficiency, accuracy and stability of the method are assessed by applying it to a number of test problems. The results are compared with the existing closed-form solutions; the scheme demonstrates that the numerical outcomes are reliable and quite accurate when contrasted with the analytical solutions and an existing numerical method. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a finite element collocation approach using cubic B-splines is employed for the numerical solution of a generalized form of the nonlinear Klein–Gordon equation. The efficiency of the method is tested on a number of examples that represent special cases of the extended equation including the sine-Gordon equation. The numerical results are compared with existing numerical and analytic solutions and the outcomes confirm that the scheme yields accurate and reliable results even when few nodes are used at the time levels. [Copyright &y& Elsevier]

In this paper, a Laplace transform decomposition numerical algorithm is introduced for solving Bratu’s problem. The numerical scheme is based on the application of Laplace transform integral operator to the differential equation. The nonlinear term is then decomposed and an iterative algorithm is constructed for the determination of the infinite series solution. The technique is illustrated with two numerical examples and the results show that the method converges rapidly and approximates the exact solution very accurately using only few iterates of the recursive scheme. [Copyright &y& Elsevier]

In the present paper, a numerical algorithm is implemented for the order verification of the asymptotic expansion solution of a second-order nonlinear differential equation arising in general relativity. The method is well-adapted and illustrated for a general class of nonlinear perturbation problems. The numerical scheme can be applied to asymptotic problems as long as the numerical solution is also available. [Copyright &y& Elsevier]

In the present paper, we derive the biorthogonality condition for Stokes flow in wedge-shaped trenches bounded by radial lines and concentric circles centered at the vertex of the wedge. This condition, is a property satisfied by the eigenfunctions and adjoint eigenfunctions needed to compute the coefficients of the eigenfunction expansion solution of the corresponding Stokes flow problem. [Copyright &y& Elsevier]

In this paper, a decomposition algorithm is presented for the solution of the coupled integral H-like equations of Chandrasekhar that arise in transport theory. An important feature of this numerical technique is its rapid convergence and the high degree of accuracy by which it approximates a solution using only few terms of its iterative scheme. [Copyright &y& Elsevier]