Suheil A. Khuri

A new approach to Bratu's problem

In this paper, a Laplace transform decomposition numerical algorithm is introduced for solving Bratus 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.

A novel approach for the solution of a class of singular boundary value problems arising in physiology

A new approach implementing a modified decomposition method in combination with the cubic B-spline collocation technique is introduced for the numerical solution of a class of singular boundary value problems arising in physiology. The domain of the problem is split into two subintervals; a modified decomposition procedure based on a special integral operator is implemented in the vicinity of the singular point and outside this domain the resulting boundary value problem is tackled by applying the B-spline scheme. Performance of this method is examined numerically; the examples reveal that the current approach converges to the exact solution rapidly and with O(h^2) accuracy. Results show that the method yields a numerical solution in very good agreement with the existing exact and approximate solutions.

Biorthogonal series solution of Stokes flow problems in sectorial regions

In this paper we develop an eigenfunction expansion method for solving Stokes flow problems in sectorial cavities that arise in fluid dynamics. The method leads to the development of a set of eigenfunctions, adjoint eigenfunctions, biorthogonality conditions, and an algorithm for the computation of the coefficients of the eigenfunction expansion. The resulting infinite system of linear equations is then solved by truncation. These biorthogonality conditions are properties satisfied by the eigenfunctions and adjoint eigenfunctions, which are used to compute the coefficients of the eigenfunction expansion solution. The biorthogonality conditions are derived for a class of fourth-order boundary value problems with variable coefficients that arise from separating variables of the governing Stokes equation. The method is applied to the slow, steady flow in a two-dimensional sectorial cavity; the fluid is set into motion by the uniform translation of a covering plate or belt.

On univalent solutions of the biharmonic equation

We analyze the univalence of the solutions of the biharmonic equation. In particular, we show that if is a biharmonic map in the form,, where is harmonic, then is starlike whenever is starlike. In addition, when,, where and are harmonic, we show that is locally univalent whenever is starlike and is orientation preserving.

On some properties of solutions of the biharmonic equation

In the present paper, the properties of the linear complex operator L(f) = Zfz - Z-fz-, which is defined on the class of complex-valued C1 functions in the plane, are investigated. It is shown that harmonicity and biharmonicity are invariant under the linear operatorL. Results concerning starlikeness and convexity of biharmonic functions versus the corresponding harmonic functions are considered. The operator L can be manipulated to express the conditions in the definitions of starlikeness and convexity in a convenient way.

Troesch’s problem: A B-spline collocation approach

Abstract A finite-element approach, based on cubic B-spline collocation, is presented for the numerical solution of Troesch’s problem. The method is used on both a uniform mesh and a piecewise-uniform Shishkin mesh, depending on the magnitude of the eigenvalues. This is due to the existence of a boundary layer at the right endpoint of the domain for relatively large eigenvalues. The problem is also solved using an adaptive spline collocation approach over a non-uniform mesh via exploiting an iterative scheme arising from Newton’s method. The convergence analysis is discussed and is shown to depend on the eigenvalues; in particular, the rate of convergence is calculated using the double-mesh principle. To demonstrate the efficiency of the method, a number of special cases are considered. The numerical solutions are compared with both the analytical solutions and other existing numerical solutions in the literature. It is observed that the results obtained by this method are quite satisfactory and accurate, and the method is applicable for a wide range of cases when contrasted with other available solutions.

A spline collocation approach for the numerical solution of a generalized nonlinear Klein-Gordon equation

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.

A Laplace variational iteration strategy for the solution of differential equations

Abstract The aim of this article is to introduce a novel Laplace variational numerical scheme, based on the variational iteration method (VIM) and Laplace transform, for the solution of certain classes of linear and nonlinear differential equations. The strategy is outlined and then illustrated through a number of test examples. The results assert that this alternative approach yields accurate results, converges rapidly and handles impulse functions and the ones with discontinuities.

A numerical approach for solving an extended Fisher-Kolomogrov-Petrovskii-Piskunov equation

In the present paper a numerical method, based on finite differences and spline collocation, is presented for the numerical solution of a generalized Fisher integro-differential equation. A composite weighted trapezoidal rule is manipulated to handle the numerical integrations which results in a closed-form difference scheme. A number of test examples are solved to assess the accuracy of the method. The numerical solutions obtained, indicate that the approach is reliable and yields results compatible with the exact solutions and consistent with other existing numerical methods. Convergence and stability of the scheme have also been discussed.

A novel fixed point iteration method for the solution of third order boundary value problems

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 Greens functions into well-known fixed point iterations, including Picards and Krasnoselskii-Manns 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.