Homotopy analysis method for fractional IVPs

In this paper, the homotopy analysis method is applied to solve linear and nonlinear fractional initial-value problems (fIVPs). The fractional derivatives are described by Caputo’s sense. Exact and/or approximate analytical solutions of the fIVPs are obtained. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the approach.     

A matrix method for fractional Sturm-Liouville problems on bounded domain

A matrix method for the solution of direct fractional Sturm-Liouville problems (SLPs) on bounded domains is proposed where the fractional derivative is defined in the Riesz sense. The scheme is based on the application of the Galerkin spectral method of orthogonal polynomials. The order of convergence of the eigenvalue approximations with respect to the matrix size is studied. Some numerical examples that confirm the theory and prove the competitiveness of the approach are finally presented.     

Homotopy perturbation method for fractional KdV-Burgers equation

In the paper, we extend the homotopy perturbation method to solve nonlinear fractional partial differential equations. The time- and space-fractional KdV-Burgers equations with initial conditions are chosen to illustrate our method. As a result, we successfully obtain some available approximate solutions of them. The results reveal that the proposed method is very effective and simple for obtaining approximate solutions of nonlinear fractional partial differential equations.     

Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation

In this paper, we adopt the homotopy analysis method (HAM) to obtain solutions of linear and nonlinear fractional diffusion and wave equation. The fractional derivative is described in the Caputo sense. Some illustrative examples are presented.     

Homotopy perturbation method for fractional KdV equation

In this paper, the homotopy perturbation method is directly applied to derive approximate solutions of the fractional KdV equation. The results reveal that the proposed method is very effective and simple for solving approximate solutions of fractional differential equations.     

Half-range solutions of indefinite Sturm-Liouville problems

For diffusion equations of indefinite Sturm-Liouville type, we develop two equivalent methods of constructing explicit representations for the solutions. The first method is based on an eigenfunction expansion whereas the second uses a Wiener-Hopf-type integral equation and factorization. Some illustrative examples are worked out.     

Broyden method for inverse non-symmetric Sturm-Liouville problems

In this paper, we propose a derivative-free method for recovering symmetric and non-symmetric potential functions of inverse Sturm-Liouville problems from the knowledge of eigenvalues. A class of boundary value methods obtained as an extension of Numerov’s method is the major tool for approximating the eigenvalues in each Broyden iteration step. Numerical examples demonstrate that the method is able to reduce the number of iteration steps, in particular for non-symmetric potentials, without accuracy loss.     

Boundary value method for inverse Sturm-Liouville problems

In this paper we present a method to recover symmetric and non-symmetric potential functions of inverse Sturm-Liouville problems from the knowledge of eigenvalues. The linear multistep method coupled with suitable boundary conditions known as boundary value method (BVM) is the main tool to approximate the eigenvalues in each iteration step of the used Newton method. The BVM was extended to work for Neumann-Neumann boundary conditions. Moreover, a suitable approximation for the asymptotic correction of the eigenvalues is given. Numerical results demonstrate that the method is able to give good results for both symmetric and non-symmetric potentials.     

Accurate solutions of fourth order Sturm-Liouville problems

Recently we introduced a new method which we call the Extended Sampling Method to compute the eigenvalues of second order Sturm-Liouville problems with eigenvalue dependent potential. We shall see in this paper how we use this method to compute the eigenvalues of fourth order Sturm-Liouville problems and present its practical use on a few examples.     

Nontrivial solutions of singular superlinear Sturm-Liouville problems

The singular superlinear Sturm-Liouville problems

     

Nontrivial solutions of singular sublinear Sturm-Liouville problems

The singular sublinear Sturm-Liouville problems

     

Homotopy perturbation method for linear programming problems

In this paper, He’s homotopy perturbation method (HPM) is applied for solving linear programming (LP) problems. This paper shows that some recent findings about this topic cannot be applied for all cases. Furthermore, we provide the correct application of HPM for LP problems. The proposed method has a simple and graceful structure. Finally, a numerical example is displayed to illustrate the proposed method.     

Existence of positive solutions for Sturm-Liouville boundary value problems on the half-line

In this paper, we establish sufficient conditions to guarantee the existence of at least one positive solution, a unique positive solution, and multiple positive solutions for the Sturm-Liouville boundary value problem on the half-line. By using an effective operator, the fixed point theorems in cone, especially Krasnoselskii fixed point theorem, can be applied to such systems and then existence criteria are established. The interesting point of the results is that the nonlinear term f can be sign-changing.     

Positive solutions of fourth-order nonlinear singular Sturm-Liouville eigenvalue problems

We consider the existence of positive solutions for the following fourth-order singular Sturm-Liouville eigenvalue problems

     

The method of external excitation for solving generalized Sturm-Liouville problems

A new numerical technique for solving the generalized Sturm-Liouville problem , b_l[w(0),λ]=b_r[w(1),λ]=0 is presented. In particular, we consider the problems when the coefficient q(x,λ) or the boundary conditions depend on the spectral parameter λ in an arbitrary nonlinear manner. The method presented is based on mathematically modelling the physical response of a system to excitation over a range of frequencies. The response amplitudes are then used to determine the eigenvalues.The results of the numerical experiments justifying the method are presented.