Qingbiao Wu

Homotopy perturbation transform method for nonlinear equations using He's polynomials

In this paper, a combined form of the Laplace transform method with the homotopy perturbation method is proposed to solve nonlinear equations. This method is called the homotopy perturbation transform method (HPTM). The nonlinear terms can be easily handled by the use of Hes polynomials. The proposed scheme finds the solution without any discretization or restrictive assumptions and avoids the round-off errors. The fact that the proposed technique solves nonlinear problems without using Adomians polynomials can be considered as a clear advantage of this algorithm over the decomposition method.

A new family of eighth-order iterative methods for solving nonlinear equations

A family of eighth-order iterative methods with four evaluations for the solution of nonlinear equations is presented. Kung and Traub conjectured that an iteration method without memory based on n evaluations could achieve optimal convergence order 2^n^-^1. The new family of eighth-order methods agrees with the conjecture of Kung-Traub for the case n=4. Therefore this family of methods has efficiency index equal to 1.682. Numerical comparisons are made with several other existing methods to show the performance of the presented methods.

The effects of variable viscosity and thermal conductivity on a thin film flow over a shrinking/stretching sheet

The effects of variable viscosity and thermal conductivity on the flow and heat transfer in a laminar liquid film on a horizontal shrinking/stretching sheet are analyzed. The similarity transformation reduces the time independent boundary layer equations for momentum and thermal energy into a set of coupled ordinary differential equations. The resulting five-parameter problem is solved by the homotopy perturbation method. The results are presented graphically to interpret various physical parameters appearing in the problem.

A CLASS OF TWO-STEP STEFFENSEN TYPE METHODS WITH FOURTH-ORDER CONVERGENCE

Abstract Based on Steffensen’s method, we derive a one-parameter class of fourth-order methods for solving nonlinear equations. In the proposed methods, an interpolating polynomial is used to get a better approximation to the derivative of the given function. Each member of the class requires three evaluations of the given function per iteration. Therefore, this class of methods has efficiency index which equals 1.587. Kung and Traub conjectured an iteration using n evaluations of f or its derivatives without memory is of convergence order at most 2 n - 1 . The new class of fourth-order methods agrees with the conjecture of Kung–Traub for the case n = 3 . Numerical comparisons are made to show the performance of the presented methods.

New variants of Jarratt’s method with sixth-order convergence

In this paper, by using the two-variable Taylor expansion formula, we introduce some new variants of Jarratt’s method with sixth-order convergence for solving univariate nonlinear equations. The proposed methods contain some recent improvements of Jarratt’s method. Furthermore, a new variant of Jarratt’s method with sixth-order convergence for solving systems of nonlinear equations is proposed only with an additional evaluation for the involved function, and not requiring the computation of new inverse. Numerical comparisons are made to show the performance of the presented methods.

An efficient iterated method for mathematical biology model

The purpose of this study is to introduce an efficient iterated homotopy perturbation transform method (IHPTM) for solving a mathematical model of HIV infection of CD4+ T cells. The equations are Laplace transformed, and the nonlinear terms are represented by He’s polynomials. The solutions are obtained in the form of rapidly convergent series with elegantly computable terms. This approach, in contrast to classical perturbation techniques, is valid even for systems without any small/large parameters and therefore can be applied more widely than traditional perturbation techniques, especially when there do not exist any small/large quantities. A good agreement of the novel method solution with the existing solutions is presented graphically and in tabulated forms to study the efficiency and accuracy of IHPTM. This study demonstrates the general validity and the great potential of the IHPTM for solving strongly nonlinear problems.

Third-order convergence theorem by using majorizing function for a modified Newton method in Banach space☆

Abstract Following the ideas of Frontini and Sormani, we present a modified Newton method in Banach space which is used to solve the nonlinear operator equation. We establish the Newton–Kantorovich convergence theorem for the modified Newton method with third-order convergence in Banach space by using majorizing function. We also get the error estimate. Finally, two examples are provided to show the application of our theorem.

Fractional variational iteration method for fractional initial-boundary value problems arising in the application of nonlinear science

In this paper, we suggest a fractional functional for the variational iteration method to solve the linear and nonlinear fractional order partial differential equations with fractional order initial and boundary conditions by using the modified Riemann-Liouville fractional derivative proposed by G. Jumarie. Fractional order Lagrange multiplier has been considered. Solution has been plotted for different values of @a.

A new fractional analytical approach via a modified Riemann–Liouville derivative

Abstract This work suggests a new analytical technique called the fractional homotopy perturbation method (FHPM) for solving fractional differential equations of any fractional order. This method is based on He’s homotopy perturbation method and the modified Riemann–Liouville derivative. The fractional differential equations are described in Jumarie’s sense. The results from introducing a modified Riemann–Liouville derivative in the cases studied show the high accuracy, simplicity and efficiency of the approach.

New family of seventh-order methods for nonlinear equations

Abstract A family of seventh-order iterative methods for the solution of nonlinear equations is presented. The new methods are based on King’s fourth-order methods and without using the second derivatives. Per iteration the new methods require three evaluations of the function and one evaluation of its first derivative. Therefore, this family of methods has efficiency index equal to 1.627. Numerical comparisons are made with several other existing methods to show the performance of the presented methods.