Shaher Momani | The University Of Jordan (original) (raw)
Papers by Shaher Momani
In this paper, we develop and analyze the use of the Optimal Homotopy Asymptotic Method (OHAM) to... more In this paper, we develop and analyze the use of the Optimal Homotopy Asymptotic Method (OHAM) to find the approximate analytical solution for partial differential equation involving fuzzy heat equation. OHAM allows the solution of the partial differential equation to be calculated in the form of an infinite series in which the components can be easily computed. The convergence theorem of this method in fuzzy case is presented and proved. This method provides us with a convenient way to control the convergence of approximation series. The method is tested on linear fuzzy heat equations and comparing the exact solution that were made with numerical results showed the effectiveness and accuracy of this method.
In this paper, the continuous genetic algorithm is applied for the solution of singular two-point... more In this paper, the continuous genetic algorithm is applied for the solution of singular two-point boundary value problems, where smooth solution curves are used throughout the evolution of the algorithm to obtain the required nodal values. The proposed technique might be considered as a variation of the finite difference method in the sense that each of the derivatives is replaced by an appropriate difference quotient approximation.
Hindawi Publishing Corporation, 2012
This paper investigates the numerical solution of nonlinear Fredholm-Volterra integro-differentia... more This paper investigates the numerical solution of nonlinear Fredholm-Volterra integro-differential equations using reproducing kernel Hilbert space method. The solution u x is represented in the form of series in the reproducing kernel space. In the mean time, the n-term approximate solution u n x is obtained and it is proved to converge to the exact solution u x . Furthermore, the proposed method has an advantage that it is possible to pick any point in the interval of integration and as well the approximate solution and its derivative will be applicable. Numerical examples are included to demonstrate the accuracy and applicability of the presented technique. The results reveal that the method is very effective and simple.
This paper investigates the numerical solution of nonlinear Fredholm-Volterra integro-differentia... more This paper investigates the numerical solution of nonlinear Fredholm-Volterra integro-differential equations using reproducing kernel Hilbert space method. The solution 𝑢 (𝑥) is represented in the form of series in the reproducing kernel space. In the mean time, the n-term approximate solution 𝑢𝑛 (𝑥) is obtained and it is proved to converge to the exact solution 𝑢 (𝑥).
Nonlinear Analysis-theory Methods & Applications, 2008
... Article Outline. 1. Introduction 2. The main theorems References. ... 2. The main theorems. I... more ... Article Outline. 1. Introduction 2. The main theorems References. ... 2. The main theorems. In this section, we shall prove our main results, we begin by proving a theorem concerned with the existence and uniqueness of mild solution for the semilinear initial value problem (1). ...
Journal of Computational and Applied Mathematics, 2008
In this paper, we obtain the series solutions to three well known equations arising in different ... more In this paper, we obtain the series solutions to three well known equations arising in different fields of science. We apply homotopy perturbation method (HPM) to Burgers equation, the regularized long wave equation and the modified Korteweg-de Vries equation. In each problem, applying HPM, we obtain the Taylor expansion of the exact solution which are convergent in desired domains.
Communications in Nonlinear Science and Numerical Simulation, 2009
In this paper, the homotopy analysis method is applied to solve linear and nonlinear fractional i... more 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.
Chaos Solitons & Fractals, 2006
(1.1) View the MathML source where ε, ν, η are parameters and α and β are parameters describing t... more (1.1) View the MathML source where ε, ν, η are parameters and α and β are parameters describing the order of the fractional time-and space-derivatives, respectively. The function u(x, t) is assumed to be a causal function of time and space, ie, vanishing for t < 0 and x < 0. The fractional ...
Applied Mathematics and Computation, 2005
The Adomian decomposition method is used to obtain analytic and approximate solutions of the spac... more The Adomian decomposition method is used to obtain analytic and approximate solutions of the space-and time-fractional telegraph equations. The space-and timefractional derivatives are considered in the Caputo sense. The analytic solutions are calculated in the form of series with easily computable terms. Some examples are given. The results reveal that the Adomian method is very effective and convenient.
Mathematics and Computers in Simulation, 2005
In this paper, a fractional Korteweg-de Vries equation (KdV for short) with initial condition is ... more In this paper, a fractional Korteweg-de Vries equation (KdV for short) with initial condition is introduced by replacing the first order time and space derivatives by fractional derivatives of order α and β with 0 < α, β ≤ 1, respectively. The fractional derivatives are described in the Caputo sense. The application of Adomian decomposition method, developed for differential equations of integer order, is extended to derive explicit and numerical solutions of the fractional KdV equation. The solutions of our model equation are calculated in the form of convergent series with easily computable components.
Applied Mathematics and Computation, 2005
The partial differential equation of diffusion is generalized by replacing the first order time d... more The partial differential equation of diffusion is generalized by replacing the first order time derivative by a fractional derivative of order a, 0 < a 6 2. An approximate solution based on the decomposition method is given for the generalized fractional diffusion (diffusion-wave) equation. The fractional derivative is described in the Caputo sense. Numerical example is given to show the application of the present technique. Results show the transition from a pure diffusion process (a = 1) to a pure wave process (a = 2).
Applied Mathematics and Computation, 2005
In this paper we use Adomian decomposition method to solve systems of nonlinear fractional differ... more In this paper we use Adomian decomposition method to solve systems of nonlinear fractional differential equations and a linear multi-term fractional differential equation by reducing it to a system of fractional equations each of order at most unity. We begin by showing how the decomposition method applies to a class of nonlinear fractional differential equations and give two examples to illustrate the efficiency of the method. Moreover, we show how the method can be applied to a general linear multi-term equation and solve several applied problems.
Journal of Computational and Applied Mathematics, 2008
In this article, a novel numerical method is proposed for nonlinear partial differential equation... more In this article, a novel numerical method is proposed for nonlinear partial differential equations with space-and time-fractional derivatives. This method is based on the two-dimensional differential transform method (DTM) and generalized Taylor's formula. The fractional derivatives are considered in the Caputo sense. Several illustrative examples are given to demonstrate the effectiveness of the present method. Results obtained using the scheme presented here agree well with the analytical solutions and the numerical results presented elsewhere. Results also show that the numerical scheme is very effective and convenient for solving nonlinear partial differential equations of fractional order.
Chaos Solitons & Fractals, 2006
In th is p ap er , He's v ariatio n al iteratio n meth o d (VIM ) is u s ed to an aly ze th e d e... more In th is p ap er , He's v ariatio n al iteratio n meth o d (VIM ) is u s ed to an aly ze th e d eflectio n o f polysilicon diaphragm of Micro Electro Mechanical Systems (MEMS) capacitive micro phone. The residual stresses in the material used to make th e diaphragm change the vibrational characteristics of the microphone diaphragm and consequently influence the microphone's first resonant frequency, cu to ff freq u en cy an d s en s itiv ity .Th e mos t s u cces s fu l dev ices u s e p o ly s ilico n as a d iap h rag m material, b ecause of its residual stress is controllable by high temperature annealing after ion implantatio n b y boron or phosphorous. External aco u s tic force causes to deflect the diaphragm of the structure and VIM is a powerful analytical method to predict the s tructural behavior and the microphone performance. Comparison of this new method with the previous app roximate solution [1], is applied to assure us about the accuracy of solution.
Physics Letters A, 2007
This Letter applies the modified He's homotopy perturbation method (HPM) suggested by Momani and ... more This Letter applies the modified He's homotopy perturbation method (HPM) suggested by Momani and Odibat to obtaining solutions of linear and nonlinear fractional diffusion and wave equations. The fractional derivative is described in the Caputo sense. Some illustrative examples are given, revealing the effectiveness and convenience of the method.
Physics Letters A, 2007
The aim of this Letter is to present an efficient and reliable treatment of the homotopy perturba... more The aim of this Letter is to present an efficient and reliable treatment of the homotopy perturbation method (HPM) for nonlinear partial differential equations with fractional time derivative. The fractional derivative is described in the Caputo sense. The modified ...
Chaos Solitons & Fractals, 2008
In this paper, a modification of He's homotopy perturbation method is presented. The new modifica... more In this paper, a modification of He's homotopy perturbation method is presented. The new modification extends the application of the method to solve nonlinear differential equations of fractional order. In this method, which does not require a small parameter in an equation, a homotopy with an imbedding parameter p 2 [0, 1] is constructed. The proposed algorithm is applied to the quadratic Riccati differential equation of fractional order. The results reveal that the method is very effective and convenient for solving nonlinear differential equations of fractional order.
International Journal of Mathematics and Mathematical Sciences, 2004
Lyapunov stability and asymptotic stability conditions for the solutions of the fractional integr... more Lyapunov stability and asymptotic stability conditions for the solutions of the fractional integrodiffrential equations x (α)
In this paper, we develop and analyze the use of the Optimal Homotopy Asymptotic Method (OHAM) to... more In this paper, we develop and analyze the use of the Optimal Homotopy Asymptotic Method (OHAM) to find the approximate analytical solution for partial differential equation involving fuzzy heat equation. OHAM allows the solution of the partial differential equation to be calculated in the form of an infinite series in which the components can be easily computed. The convergence theorem of this method in fuzzy case is presented and proved. This method provides us with a convenient way to control the convergence of approximation series. The method is tested on linear fuzzy heat equations and comparing the exact solution that were made with numerical results showed the effectiveness and accuracy of this method.
In this paper, the continuous genetic algorithm is applied for the solution of singular two-point... more In this paper, the continuous genetic algorithm is applied for the solution of singular two-point boundary value problems, where smooth solution curves are used throughout the evolution of the algorithm to obtain the required nodal values. The proposed technique might be considered as a variation of the finite difference method in the sense that each of the derivatives is replaced by an appropriate difference quotient approximation.
Hindawi Publishing Corporation, 2012
This paper investigates the numerical solution of nonlinear Fredholm-Volterra integro-differentia... more This paper investigates the numerical solution of nonlinear Fredholm-Volterra integro-differential equations using reproducing kernel Hilbert space method. The solution u x is represented in the form of series in the reproducing kernel space. In the mean time, the n-term approximate solution u n x is obtained and it is proved to converge to the exact solution u x . Furthermore, the proposed method has an advantage that it is possible to pick any point in the interval of integration and as well the approximate solution and its derivative will be applicable. Numerical examples are included to demonstrate the accuracy and applicability of the presented technique. The results reveal that the method is very effective and simple.
This paper investigates the numerical solution of nonlinear Fredholm-Volterra integro-differentia... more This paper investigates the numerical solution of nonlinear Fredholm-Volterra integro-differential equations using reproducing kernel Hilbert space method. The solution 𝑢 (𝑥) is represented in the form of series in the reproducing kernel space. In the mean time, the n-term approximate solution 𝑢𝑛 (𝑥) is obtained and it is proved to converge to the exact solution 𝑢 (𝑥).
Nonlinear Analysis-theory Methods & Applications, 2008
... Article Outline. 1. Introduction 2. The main theorems References. ... 2. The main theorems. I... more ... Article Outline. 1. Introduction 2. The main theorems References. ... 2. The main theorems. In this section, we shall prove our main results, we begin by proving a theorem concerned with the existence and uniqueness of mild solution for the semilinear initial value problem (1). ...
Journal of Computational and Applied Mathematics, 2008
In this paper, we obtain the series solutions to three well known equations arising in different ... more In this paper, we obtain the series solutions to three well known equations arising in different fields of science. We apply homotopy perturbation method (HPM) to Burgers equation, the regularized long wave equation and the modified Korteweg-de Vries equation. In each problem, applying HPM, we obtain the Taylor expansion of the exact solution which are convergent in desired domains.
Communications in Nonlinear Science and Numerical Simulation, 2009
In this paper, the homotopy analysis method is applied to solve linear and nonlinear fractional i... more 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.
Chaos Solitons & Fractals, 2006
(1.1) View the MathML source where ε, ν, η are parameters and α and β are parameters describing t... more (1.1) View the MathML source where ε, ν, η are parameters and α and β are parameters describing the order of the fractional time-and space-derivatives, respectively. The function u(x, t) is assumed to be a causal function of time and space, ie, vanishing for t &lt; 0 and x &lt; 0. The fractional ...
Applied Mathematics and Computation, 2005
The Adomian decomposition method is used to obtain analytic and approximate solutions of the spac... more The Adomian decomposition method is used to obtain analytic and approximate solutions of the space-and time-fractional telegraph equations. The space-and timefractional derivatives are considered in the Caputo sense. The analytic solutions are calculated in the form of series with easily computable terms. Some examples are given. The results reveal that the Adomian method is very effective and convenient.
Mathematics and Computers in Simulation, 2005
In this paper, a fractional Korteweg-de Vries equation (KdV for short) with initial condition is ... more In this paper, a fractional Korteweg-de Vries equation (KdV for short) with initial condition is introduced by replacing the first order time and space derivatives by fractional derivatives of order α and β with 0 < α, β ≤ 1, respectively. The fractional derivatives are described in the Caputo sense. The application of Adomian decomposition method, developed for differential equations of integer order, is extended to derive explicit and numerical solutions of the fractional KdV equation. The solutions of our model equation are calculated in the form of convergent series with easily computable components.
Applied Mathematics and Computation, 2005
The partial differential equation of diffusion is generalized by replacing the first order time d... more The partial differential equation of diffusion is generalized by replacing the first order time derivative by a fractional derivative of order a, 0 < a 6 2. An approximate solution based on the decomposition method is given for the generalized fractional diffusion (diffusion-wave) equation. The fractional derivative is described in the Caputo sense. Numerical example is given to show the application of the present technique. Results show the transition from a pure diffusion process (a = 1) to a pure wave process (a = 2).
Applied Mathematics and Computation, 2005
In this paper we use Adomian decomposition method to solve systems of nonlinear fractional differ... more In this paper we use Adomian decomposition method to solve systems of nonlinear fractional differential equations and a linear multi-term fractional differential equation by reducing it to a system of fractional equations each of order at most unity. We begin by showing how the decomposition method applies to a class of nonlinear fractional differential equations and give two examples to illustrate the efficiency of the method. Moreover, we show how the method can be applied to a general linear multi-term equation and solve several applied problems.
Journal of Computational and Applied Mathematics, 2008
In this article, a novel numerical method is proposed for nonlinear partial differential equation... more In this article, a novel numerical method is proposed for nonlinear partial differential equations with space-and time-fractional derivatives. This method is based on the two-dimensional differential transform method (DTM) and generalized Taylor's formula. The fractional derivatives are considered in the Caputo sense. Several illustrative examples are given to demonstrate the effectiveness of the present method. Results obtained using the scheme presented here agree well with the analytical solutions and the numerical results presented elsewhere. Results also show that the numerical scheme is very effective and convenient for solving nonlinear partial differential equations of fractional order.
Chaos Solitons & Fractals, 2006
In th is p ap er , He's v ariatio n al iteratio n meth o d (VIM ) is u s ed to an aly ze th e d e... more In th is p ap er , He's v ariatio n al iteratio n meth o d (VIM ) is u s ed to an aly ze th e d eflectio n o f polysilicon diaphragm of Micro Electro Mechanical Systems (MEMS) capacitive micro phone. The residual stresses in the material used to make th e diaphragm change the vibrational characteristics of the microphone diaphragm and consequently influence the microphone's first resonant frequency, cu to ff freq u en cy an d s en s itiv ity .Th e mos t s u cces s fu l dev ices u s e p o ly s ilico n as a d iap h rag m material, b ecause of its residual stress is controllable by high temperature annealing after ion implantatio n b y boron or phosphorous. External aco u s tic force causes to deflect the diaphragm of the structure and VIM is a powerful analytical method to predict the s tructural behavior and the microphone performance. Comparison of this new method with the previous app roximate solution [1], is applied to assure us about the accuracy of solution.
Physics Letters A, 2007
This Letter applies the modified He's homotopy perturbation method (HPM) suggested by Momani and ... more This Letter applies the modified He's homotopy perturbation method (HPM) suggested by Momani and Odibat to obtaining solutions of linear and nonlinear fractional diffusion and wave equations. The fractional derivative is described in the Caputo sense. Some illustrative examples are given, revealing the effectiveness and convenience of the method.
Physics Letters A, 2007
The aim of this Letter is to present an efficient and reliable treatment of the homotopy perturba... more The aim of this Letter is to present an efficient and reliable treatment of the homotopy perturbation method (HPM) for nonlinear partial differential equations with fractional time derivative. The fractional derivative is described in the Caputo sense. The modified ...
Chaos Solitons & Fractals, 2008
In this paper, a modification of He's homotopy perturbation method is presented. The new modifica... more In this paper, a modification of He's homotopy perturbation method is presented. The new modification extends the application of the method to solve nonlinear differential equations of fractional order. In this method, which does not require a small parameter in an equation, a homotopy with an imbedding parameter p 2 [0, 1] is constructed. The proposed algorithm is applied to the quadratic Riccati differential equation of fractional order. The results reveal that the method is very effective and convenient for solving nonlinear differential equations of fractional order.
International Journal of Mathematics and Mathematical Sciences, 2004
Lyapunov stability and asymptotic stability conditions for the solutions of the fractional integr... more Lyapunov stability and asymptotic stability conditions for the solutions of the fractional integrodiffrential equations x (α)