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Iterative differential quadrature solutions for Bratu problem

2017-05-11
Ola Ragb , L.F. Seddek , M. S. Matbuly

A modified diffusion coefficient technique for the convection diffusion equation

2013-04-22
S.A. Mohamed , Nazira Mohamed , Ahmed F. Abdel Gawad +1 more

The Differential Quadrature Solution of Reaction-Diffusion Equation Using Explicit and Implicit Numerical Schemes

2014-01-01
Mohamed Salah , R. M. Amer , M. S. Matbuly
In this paper, two different numerical schemes, namely the Runge-Kutta fourth order method and the implicit Euler method with perturbation method of the second degree, are applied to solve the 
 In this paper, two different numerical schemes, namely the Runge-Kutta fourth order method and the implicit Euler method with perturbation method of the second degree, are applied to solve the nonlinear thermal wave in one and two dimensions using the differential quadrature method. The aim of this paper is to make comparison between previous numerical schemes and detect which is more efficient and more accurate by comparing the obtained results with the available analytical ones and computing the computational time.
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Vibration Analysis of Piezoelectric Composite Plate Resting on Nonlinear Elastic Foundations Using Sinc and Discrete Singular Convolution Differential Quadrature Techniques

2020-07-15
Ola Ragb , Mohamed Salah , M. S. Matbuly +1 more
In this work, free vibration of the piezoelectric composite plate resting on nonlinear elastic foundations is examined. The three-dimensionality of elasticity theory and piezoelectricity is used to derive the governing 
 In this work, free vibration of the piezoelectric composite plate resting on nonlinear elastic foundations is examined. The three-dimensionality of elasticity theory and piezoelectricity is used to derive the governing equation of motion. By implementing two differential quadrature schemes and applying different boundary conditions, the problem is converted to a nonlinear eigenvalue problem. The perturbation method and iterative quadrature formula are used to solve the obtained equation. Numerical analysis of the proposed schemes is introduced to demonstrate the accuracy and efficiency of the obtained results. The obtained results are compared with available results in the literature, showing excellent agreement. Additionally, the proposed schemes have higher efficiency than previous schemes. Furthermore, a parametric study is introduced to investigate the effect of elastic foundation parameters, different materials of sensors and actuators, and elastic and geometric characteristics of the composite plate on the natural frequencies and mode shapes.
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Fractional differential quadrature techniques for fractional order Cauchy reaction‐diffusion equations

2023-02-13
Ola Ragb , Abdul‐Majid Wazwaz , Mokhtar Mohamed +2 more
This paper aims to explore and apply differential quadrature based on different test functions to find an efficient numerical solution of fractional order Cauchy reaction‐diffusion equations (CRDEs). The governing system 
 This paper aims to explore and apply differential quadrature based on different test functions to find an efficient numerical solution of fractional order Cauchy reaction‐diffusion equations (CRDEs). The governing system is discretized through time and space via novel techniques of differential quadrature method and the fractional operator of Caputo kind. Two problems are offered to explain the accuracy of the numerical algorithms. To verify the reliability, accuracy, efficiency, and speed of these methods, computed results are compared numerically and graphically with the exact and semi‐exact solutions. Then mainly, we deal with absolute errors and L ∞ errors to study the convergence of the presented methods. For each technique, MATLAB Code is designed to solve these problems with the error reaching ≀1 × 10 −5 . In addition, a parametric analysis is presented to discuss influence of fractional order derivative on results. The achieved solutions prove the viability of the presented methods and demonstrate that these methods are easy to implement, effective, high accurate, and appropriate for studying fractional partial differential equations emerging in fields related to science and engineering.

Calculation of Four-Dimensional Unsteady Gas Flow via Different Quadrature Schemes and Runge-Kutta 4th Order Method

2023-09-05
Mohamed Salah , M. S. Matbuly , Ömer Cívalek +1 more
In this study, a (3+1) dimensional unstable gas flow system is applied and solved successfully via differential quadrature techniques based on various shape functions.The governing system of nonlinear four-dimensional unsteady 
 In this study, a (3+1) dimensional unstable gas flow system is applied and solved successfully via differential quadrature techniques based on various shape functions.The governing system of nonlinear four-dimensional unsteady Navier-Stokes equations of gas dynamics is reduced to the system of nonlinear ordinary differential equations using different quadrature techniques.Then, Runge-Kutta 4th order method is employed to solve the resulting system of equations.To obtain the solution of this equation, a MATLAB code is designed.The validity of these techniques is achieved by the comparison with the exact solution where the error reach to ≀ 1×10 -5 .Also, these solutions are discussed by seven various statistical analysis.Then, a parametric analysis is presented to discuss the effect of adiabatic index parameter on the velocity, pressure, and density profiles.From these computations, it is found that Discrete singular convolution based on Regularized Shannon kernels is a stable, efficient numerical technique and its strength has been appeared in this application.Also, this technique can be able to solve higher dimensional nonlinear problems in various regions of physical and numerical sciences.
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Modeling and Solution of Reaction–Diffusion Equations by Using the Quadrature and Singular Convolution Methods

2022-10-21
Ola Ragb , Mohamed Salah , M. S. Matbuly +2 more
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An Efficient Method to Solve Thermal Wave Equation

2014-01-01
Mohamed Salah , R. M. Amer , M. S. Matbuly
In this paper, an efficient technique of differential quadrature method and perturbation method is employed to analyze reaction-diffusion problems. An efficient method is presented to solve thermal wave propagation model 
 In this paper, an efficient technique of differential quadrature method and perturbation method is employed to analyze reaction-diffusion problems. An efficient method is presented to solve thermal wave propagation model in one and two dimensions. The proposed method marches in the time direction block by block and there are several time levels in each block. The global method of differential quadrature is applied in each block to discretize both the spatial and temporal derivatives. Furthermore, the proposed method is validated by comparing the obtained results with the available analytical ones and also compared with the hybrid technique of differential quadrature method and Runge-Kutta fourth order method.
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Variable fractional modeling of nonlinear Gilson–Pickering equation via discrete singular convolution method

2025-04-18
Ola Ragb , M. S. Matbuly , Mustafa İnç +2 more

Variable fractional modeling of nonlinear Gilson–Pickering equation via discrete singular convolution method

2025-04-18
Ola Ragb , M. S. Matbuly , Mustafa İnç +2 more

Calculation of Four-Dimensional Unsteady Gas Flow via Different Quadrature Schemes and Runge-Kutta 4th Order Method

2023-09-05
Mohamed Salah , M. S. Matbuly , Ömer Cívalek +1 more
In this study, a (3+1) dimensional unstable gas flow system is applied and solved successfully via differential quadrature techniques based on various shape functions.The governing system of nonlinear four-dimensional unsteady 
 In this study, a (3+1) dimensional unstable gas flow system is applied and solved successfully via differential quadrature techniques based on various shape functions.The governing system of nonlinear four-dimensional unsteady Navier-Stokes equations of gas dynamics is reduced to the system of nonlinear ordinary differential equations using different quadrature techniques.Then, Runge-Kutta 4th order method is employed to solve the resulting system of equations.To obtain the solution of this equation, a MATLAB code is designed.The validity of these techniques is achieved by the comparison with the exact solution where the error reach to ≀ 1×10 -5 .Also, these solutions are discussed by seven various statistical analysis.Then, a parametric analysis is presented to discuss the effect of adiabatic index parameter on the velocity, pressure, and density profiles.From these computations, it is found that Discrete singular convolution based on Regularized Shannon kernels is a stable, efficient numerical technique and its strength has been appeared in this application.Also, this technique can be able to solve higher dimensional nonlinear problems in various regions of physical and numerical sciences.
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Fractional differential quadrature techniques for fractional order Cauchy reaction‐diffusion equations

2023-02-13
Ola Ragb , Abdul‐Majid Wazwaz , Mokhtar Mohamed +2 more
This paper aims to explore and apply differential quadrature based on different test functions to find an efficient numerical solution of fractional order Cauchy reaction‐diffusion equations (CRDEs). The governing system 
 This paper aims to explore and apply differential quadrature based on different test functions to find an efficient numerical solution of fractional order Cauchy reaction‐diffusion equations (CRDEs). The governing system is discretized through time and space via novel techniques of differential quadrature method and the fractional operator of Caputo kind. Two problems are offered to explain the accuracy of the numerical algorithms. To verify the reliability, accuracy, efficiency, and speed of these methods, computed results are compared numerically and graphically with the exact and semi‐exact solutions. Then mainly, we deal with absolute errors and L ∞ errors to study the convergence of the presented methods. For each technique, MATLAB Code is designed to solve these problems with the error reaching ≀1 × 10 −5 . In addition, a parametric analysis is presented to discuss influence of fractional order derivative on results. The achieved solutions prove the viability of the presented methods and demonstrate that these methods are easy to implement, effective, high accurate, and appropriate for studying fractional partial differential equations emerging in fields related to science and engineering.

Modeling and Solution of Reaction–Diffusion Equations by Using the Quadrature and Singular Convolution Methods

2022-10-21
Ola Ragb , Mohamed Salah , M. S. Matbuly +2 more
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Vibration Analysis of Piezoelectric Composite Plate Resting on Nonlinear Elastic Foundations Using Sinc and Discrete Singular Convolution Differential Quadrature Techniques

2020-07-15
Ola Ragb , Mohamed Salah , M. S. Matbuly +1 more
In this work, free vibration of the piezoelectric composite plate resting on nonlinear elastic foundations is examined. The three-dimensionality of elasticity theory and piezoelectricity is used to derive the governing 
 In this work, free vibration of the piezoelectric composite plate resting on nonlinear elastic foundations is examined. The three-dimensionality of elasticity theory and piezoelectricity is used to derive the governing equation of motion. By implementing two differential quadrature schemes and applying different boundary conditions, the problem is converted to a nonlinear eigenvalue problem. The perturbation method and iterative quadrature formula are used to solve the obtained equation. Numerical analysis of the proposed schemes is introduced to demonstrate the accuracy and efficiency of the obtained results. The obtained results are compared with available results in the literature, showing excellent agreement. Additionally, the proposed schemes have higher efficiency than previous schemes. Furthermore, a parametric study is introduced to investigate the effect of elastic foundation parameters, different materials of sensors and actuators, and elastic and geometric characteristics of the composite plate on the natural frequencies and mode shapes.
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Iterative differential quadrature solutions for Bratu problem

2017-05-11
Ola Ragb , L.F. Seddek , M. S. Matbuly

An Efficient Method to Solve Thermal Wave Equation

2014-01-01
Mohamed Salah , R. M. Amer , M. S. Matbuly
In this paper, an efficient technique of differential quadrature method and perturbation method is employed to analyze reaction-diffusion problems. An efficient method is presented to solve thermal wave propagation model 
 In this paper, an efficient technique of differential quadrature method and perturbation method is employed to analyze reaction-diffusion problems. An efficient method is presented to solve thermal wave propagation model in one and two dimensions. The proposed method marches in the time direction block by block and there are several time levels in each block. The global method of differential quadrature is applied in each block to discretize both the spatial and temporal derivatives. Furthermore, the proposed method is validated by comparing the obtained results with the available analytical ones and also compared with the hybrid technique of differential quadrature method and Runge-Kutta fourth order method.
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The Differential Quadrature Solution of Reaction-Diffusion Equation Using Explicit and Implicit Numerical Schemes

2014-01-01
Mohamed Salah , R. M. Amer , M. S. Matbuly
In this paper, two different numerical schemes, namely the Runge-Kutta fourth order method and the implicit Euler method with perturbation method of the second degree, are applied to solve the 
 In this paper, two different numerical schemes, namely the Runge-Kutta fourth order method and the implicit Euler method with perturbation method of the second degree, are applied to solve the nonlinear thermal wave in one and two dimensions using the differential quadrature method. The aim of this paper is to make comparison between previous numerical schemes and detect which is more efficient and more accurate by comparing the obtained results with the available analytical ones and computing the computational time.
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A modified diffusion coefficient technique for the convection diffusion equation

2013-04-22
S.A. Mohamed , Nazira Mohamed , Ahmed F. Abdel Gawad +1 more
Coauthor Papers Together
Mohamed Salah 7
Ola Ragb 6
Ömer Cívalek 2
R. M. Amer 2
Mokhtar Mohamed 1
Hakan Ersoy 1
S.A. Mohamed 1
Shahram Rezapour 1
L.F. Seddek 1
Nazira Mohamed 1
Ahmed F. Abdel Gawad 1
Mustafa İnç 1
Rania B. M. Amer 1
Abdul‐Majid Wazwaz 1

He’s homotopy perturbation method: An effective tool for solving nonlinear integral and integro-differential equations

2009-04-24
Jafar Saberi‐Nadjafi , Asghar Ghorbani

A sinc-collocation method for initial value problems

1997-01-01
Timothy S. Carlson , Jack D. Dockery , John Lund
A collocation procedure is developed for the initial value problem $u'(t) = f(t,u(t))$, $u(0) = 0$, using the globally defined sinc basis functions. It is shown that this sinc procedure 
 A collocation procedure is developed for the initial value problem $u'(t) = f(t,u(t))$, $u(0) = 0$, using the globally defined sinc basis functions. It is shown that this sinc procedure converges to the solution at an exponential rate, i.e., $\mathcal { O} (M^{2} \exp (-\kappa \sqrt {M}) )$ where $\kappa > 0$ and $2M$ basis functions are used in the expansion. Problems on the domains $\mathbb {R} = (-\infty ,\infty )$ and $\mathbb {R} ^{+} = (0,\infty )$ are used to illustrate the implementation and accuracy of the procedure.
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Exponential finite difference methods for solving Newell–Whitehead–Segel equation

2020-03-02
Nayrouz Hilal , Sami Injrou , Ramez Karroum
Abstract This work presents two different finite difference methods to compute the numerical solutions for Newell–Whitehead–Segel partial differential equation, which are implicit exponential finite difference method and fully implicit exponential 
 Abstract This work presents two different finite difference methods to compute the numerical solutions for Newell–Whitehead–Segel partial differential equation, which are implicit exponential finite difference method and fully implicit exponential finite difference method. Implicit exponential methods lead to nonlinear systems. Newton method is used to solve the resulting systems. Stability and consistency are discussed. To illustrate the accuracy of the proposed numerical methods, some examples are delivered at the end.
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Polynomial differential quadrature method for numerical solutions of the generalized Fitzhugh–Nagumo equation with time-dependent coefficients

2014-07-08
Ram Jiwari , Rishabh Gupta , Vikas Kumar
In this paper, polynomial differential quadrature method (PDQM) is applied to find the numerical solution of the generalized Fitzhugh–Nagumo equation with time-dependent coefficients in one dimensional space. The PDQM reduces 
 In this paper, polynomial differential quadrature method (PDQM) is applied to find the numerical solution of the generalized Fitzhugh–Nagumo equation with time-dependent coefficients in one dimensional space. The PDQM reduces the problem into a system of first order non-linear differential equations. Then, the obtained system is solved by optimal four-stage, order three strong stability-preserving time-stepping Runge–Kutta (SSP-RK43) scheme. The accuracy and efficiency of the proposed method are demonstrated by three test examples. The numerical results are shown in max absolute errors (L∞), root mean square errors (RMS) and relative errors (L2) forms. Numerical solutions obtained by this method when compared with the exact solutions reveal that the obtained solutions are very similar to the exact ones.

Studies of the accuracy of time integration methods for reaction–diffusion equations

2003-11-24
David L. Ropp , John N. Shadid , Curtis Ober

Analytical solution for cauchy reaction-diffusion problems by homotopy perturbation method

2010-06-01
Md. Sazzad Hossien Chowdhury , Ishak Hashim
In this paper, the homotopy-perturbation method (HPM) is applied to obtain approximate analytical solutions for the Cauchy reaction-diffusion problems. HPM yields solutions in convergent series forms with easily computable terms. 
 In this paper, the homotopy-perturbation method (HPM) is applied to obtain approximate analytical solutions for the Cauchy reaction-diffusion problems. HPM yields solutions in convergent series forms with easily computable terms. The HPM is tested for several examples. Comparisons of the results obtained by the HPM with that obtained by the Adomian decomposition method (ADM), homotopy analysis method (HAM) and the exact solutions show the efficiency of HPM.

The Differential Quadrature Solution of Reaction-Diffusion Equation Using Explicit and Implicit Numerical Schemes

2014-01-01
Mohamed Salah , R. M. Amer , M. S. Matbuly
In this paper, two different numerical schemes, namely the Runge-Kutta fourth order method and the implicit Euler method with perturbation method of the second degree, are applied to solve the 
 In this paper, two different numerical schemes, namely the Runge-Kutta fourth order method and the implicit Euler method with perturbation method of the second degree, are applied to solve the nonlinear thermal wave in one and two dimensions using the differential quadrature method. The aim of this paper is to make comparison between previous numerical schemes and detect which is more efficient and more accurate by comparing the obtained results with the available analytical ones and computing the computational time.
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A comparison between the wavelet-Galerkin and the sinc-Galerkin methods in solving nonhomogeneous heat equations

2002-01-01
Mohamed El‐Gamel , Ahmed I. Zayed

Discrete singular convolution for the solution of the Fokker–Planck equation

1999-05-08
Guo-Wei Wei
This paper introduces a discrete singular convolution algorithm for solving the Fokker–Planck equation. Singular kernels of the Hilbert-type and the delta type are presented for numerical computations. Various sequences of 
 This paper introduces a discrete singular convolution algorithm for solving the Fokker–Planck equation. Singular kernels of the Hilbert-type and the delta type are presented for numerical computations. Various sequences of approximations to the singular kernels are discussed. A numerical algorithm is proposed to incorporate the approximation kernels for physical applications. Three standard problems, the Lorentz Fokker–Planck equation, the bistable model and the Henon–Heiles system, are utilized to test the accuracy, reliability, and speed of convergency of the present approach. All results are in excellent agreement with those of previous methods in the field.

Numerical method for the solution of special nonlinear fourth-order boundary value problems

2003-09-03
Mohamed El‐Gamel , S.H. Behiry , Hagar Hashish

Numerical Solution of Singularly Perturbed Differential-Difference Equations with Small Shifts of Mixed Type by Differential Quadrature Method

2012-02-01
Hari Shankar Prasad , Y. N. Reddy
In this paper, we have presented the Differential Quadrature Method (DQM) for finding the numerical solution of boundary-value problems for a singularly perturbed differential-difference equation of mixed type, i.e., containing 
 In this paper, we have presented the Differential Quadrature Method (DQM) for finding the numerical solution of boundary-value problems for a singularly perturbed differential-difference equation of mixed type, i.e., containing both terms having a negative shift and terms having a positive shift. Such problems are associated with expected first exit time problems of the membrane potential in models for the neuron. The Differential Quadrature Method is an efficient descritization technique in solving initial and/or boundary value problems accurately using a considerably small number of grid points. To demonstrate the applicability of the method, we have solved the model examples and compared the computational results with the exact solutions. Comparisons showed that the method is capable of achieving high accuracy and efficiency.

A highly accurate method to solve Fisher’s equation

2012-02-28
Mehdi Bastani , Davod Khojasteh Salkuyeh

Iterative differential quadrature solutions for Bratu problem

2017-05-11
Ola Ragb , L.F. Seddek , M. S. Matbuly

Perturbative linearization of reaction diffusion equations

2003-02-12
Sanjay Puri , Kay Jörg Wiese
We develop perturbative expansions to obtain solutions for the initial-value problems of two important reaction–diffusion systems, namely the Fisher equation and the time-dependent Ginzburg–Landau equation. The starting point of our 
 We develop perturbative expansions to obtain solutions for the initial-value problems of two important reaction–diffusion systems, namely the Fisher equation and the time-dependent Ginzburg–Landau equation. The starting point of our expansion is the corresponding singular-perturbation solution. This approach transforms the solution of nonlinear reaction–diffusion equations into the solution of a hierarchy of linear equations. Our numerical results demonstrate that this hierarchy rapidly converges to the exact solution.
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Finite-Difference Schemes for Reaction–Diffusion Equations Modeling Predator–Prey Interactions in MATLAB

2007-02-01
Marcus R. Garvie

Solution of Density Dependent Nonlinear Reaction-Diffusion Equation Using Differential Quadrature Method

2010-05-28
GĂŒlnihal Meral
In this study, the density dependent nonlinear reactiondiffusion equation, which arises in the insect dispersal models, is solved using the combined application of differential quadrature method(DQM) and implicit Euler method. 
 In this study, the density dependent nonlinear reactiondiffusion equation, which arises in the insect dispersal models, is solved using the combined application of differential quadrature method(DQM) and implicit Euler method. The polynomial based DQM is used to discretize the spatial derivatives of the problem. The resulting time-dependent nonlinear system of ordinary differential equations(ODE’s) is solved by using implicit Euler method. The computations are carried out for a Cauchy problem defined by a onedimensional density dependent nonlinear reaction-diffusion equation which has an exact solution. The DQM solution is found to be in a very good agreement with the exact solution in terms of maximum absolute error. The DQM solution exhibits superior accuracy at large time levels tending to steady-state. Furthermore, using an implicit method in the solution procedure leads to stable solutions and larger time steps could be used. Keywords—Density Dependent Nonlinear Reaction-Diffusion Equation, Differential Quadrature Method, Implicit Euler Method.

Vibration Analysis of Piezoelectric Composite Plate Resting on Nonlinear Elastic Foundations Using Sinc and Discrete Singular Convolution Differential Quadrature Techniques

2020-07-15
Ola Ragb , Mohamed Salah , M. S. Matbuly +1 more
In this work, free vibration of the piezoelectric composite plate resting on nonlinear elastic foundations is examined. The three-dimensionality of elasticity theory and piezoelectricity is used to derive the governing 
 In this work, free vibration of the piezoelectric composite plate resting on nonlinear elastic foundations is examined. The three-dimensionality of elasticity theory and piezoelectricity is used to derive the governing equation of motion. By implementing two differential quadrature schemes and applying different boundary conditions, the problem is converted to a nonlinear eigenvalue problem. The perturbation method and iterative quadrature formula are used to solve the obtained equation. Numerical analysis of the proposed schemes is introduced to demonstrate the accuracy and efficiency of the obtained results. The obtained results are compared with available results in the literature, showing excellent agreement. Additionally, the proposed schemes have higher efficiency than previous schemes. Furthermore, a parametric study is introduced to investigate the effect of elastic foundation parameters, different materials of sensors and actuators, and elastic and geometric characteristics of the composite plate on the natural frequencies and mode shapes.
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ON THE FINITE ELEMENT APPROXIMATION OF SYSTEMS OF REACTION-DIFFUSION EQUATIONS BY p/hp METHODS

2010-01-01
Christos , Xenophontos , Lisa Lisa Lisa +1 more

New exact solutions for a reaction-diffusion equation and a quasi-Camassa Holm equation.

2004-01-01
Jianqin Mei , Hongqing Zhang , Dongmei Jiang
In this paper, the projective Riccati equation method, presented by Yan in (1), is used for obtaining exact solutions, solitary solutions as well as periodic solutions of the reaction-diffusion equation 
 In this paper, the projective Riccati equation method, presented by Yan in (1), is used for obtaining exact solutions, solitary solutions as well as periodic solutions of the reaction-diffusion equation and the quasi-Camassa Holm equation.

A Differential Quadrature as a numerical method to solve differential equations

1999-09-23
Tong Wu , G. R. Liu

The Legendre wavelet method for solving initial value problems of Bratu-type

2012-01-20
S. Venkatesh , S. K. Ayyaswamy , S. Raja Balachandar

A new operational matrix for solving fractional-order differential equations

2009-07-23
Abbas Saadatmandi , Mehdi Dehghan

Solitary wave and multi-front wave collisions for the Bogoyavlenskii–Kadomtsev–Petviashili equation in physics, biology and electrical networks

2015-11-05
Xi-Yang Xie , Bo Tian , Wen‐Rong Sun +2 more
In this paper, we investigate a Bogoyavlenskii–Kadomtsev–Petviashili equation, which can be used to describe the propagation of nonlinear waves in physics, biology and electrical networks. We find that the equation 
 In this paper, we investigate a Bogoyavlenskii–Kadomtsev–Petviashili equation, which can be used to describe the propagation of nonlinear waves in physics, biology and electrical networks. We find that the equation is PainlevĂ© integrable. With symbolic computation, Hirota bilinear forms, solitary waves and multi-front waves are derived. Elastic collisions between/among the two and three solitary waves are graphically discussed, where the waves maintain their shapes, amplitudes and velocities after the collision only with some phase shifts. Inelastic collisions among the multi-front waves are discussed, where the front waves coalesce into one larger front wave in their collision region.

Commutators of fractional integral operators on Vanishing-Morrey spaces

2007-07-05
Maria Alessandra Ragusa

Velocity of Radial Expansion of Contrast-enhancing Gliomas and the Effectiveness of Radiotherapy in Individual Patients: a Proof of Principle

2008-03-07
Kristin R. Swanson , Hana L.P. Harpold , Danielle L. Peacock +6 more

High Accuracy Iterative Solution of Convection Diffusion Equation with Boundary Layers on Nonuniform Grids

2001-08-01
Lixin Ge , Jun Zhang

Variational iteration method: Green’s functions and fixed point iterations perspective

2014-02-20
Sami Khuri , A. Sayfy
The aim of this article is to demonstrate that the variational iteration method "VIM" is in many instances a version of fixed point iteration methods such as Picard's scheme. In 
 The aim of this article is to demonstrate that the variational iteration method "VIM" is in many instances a version of fixed point iteration methods such as Picard's scheme. In a wide range of problems, the correction functional resulting from the VIM can be interpreted and/or formulated from well-known fixed point strategies using Green's functions. A number of examples are included to assert the validity of our claim. The test problems include first and higher order initial value problems.

Upwind local differential quadrature method for solving incompressible viscous flow

2000-07-01
Jianan Sun , Zhu Zheng-you

Fractional Calculus in Bioengineering

2007-02-12
Bruce J. West

Gaussian waves in the Fitzhugh–Nagumo equation demonstrate one role of the auxiliary function H(x,t) in the homotopy analysis method

2011-08-03
Robert A. Van Gorder

On the integral solution of the one-dimensional Bratu problem

2013-03-26
A. Mohsen

Iterative process acceleration of calculation of unsteady, viscous, compressible, and heat‐conductive gas flows

2014-11-17
Kiril S. Shterev
SUMMARY In this paper, extrapolation technique is introduced in the Semi‐Implicit Method for Pressure‐Linked Equations ‐ Time Step (SIMPLE‐TS) finite volume iterative algorithm for calculation of compressible Navier–Stokes–Fourier equations subject 
 SUMMARY In this paper, extrapolation technique is introduced in the Semi‐Implicit Method for Pressure‐Linked Equations ‐ Time Step (SIMPLE‐TS) finite volume iterative algorithm for calculation of compressible Navier–Stokes–Fourier equations subject of slip and jump boundary conditions. The initial state, required by the iterative solver in simulation of unsteady flow problems, is approximated in time by Lagrange polynomial extrapolation in each node. The approach is applicable to a parallel code in a straightforward way due to algorithmic independence of the neighboring nodes in the computational grid. A criterion is proposed to determine the order of extrapolation polynomial and stop the extrapolation execution, when the local steady state is reached. The approach is tested on different microflow problems: Couette flow, flow past a square in a microchannel at subsonic and supersonic speeds, and convective Rayleigh–BĂ©nard flow of a rarefied gas. The acceleration varies from 1.14‐fold to 2.8‐fold. Copyright © 2014 John Wiley & Sons, Ltd.
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Theory and Applications of Fractional Differential Equations

2006-01-01
Anatoly A. Kilbas , H. M. Srivastava , Juan J. Trujillo

Fractional derivatives in complex planes

2009-01-19
Changpin Li , Xuanhung Dao , Peng Guo

Numerical Methods for the Variable-Order Fractional Advection-Diffusion Equation with a Nonlinear Source Term

2009-01-01
Peixian Zhuang , F. Liu , Vo Anh +1 more
In this paper, we consider a variable-order fractional advection-diffusion equation with a nonlinear source term on a finite domain. Explicit and implicit Euler approximations for the equation are proposed. Stability 
 In this paper, we consider a variable-order fractional advection-diffusion equation with a nonlinear source term on a finite domain. Explicit and implicit Euler approximations for the equation are proposed. Stability and convergence of the methods are discussed. Moveover, we also present a fractional method of lines, a matrix transfer technique, and an extrapolation method for the equation. Some numerical examples are given, and the results demonstrate the effectiveness of theoretical analysis.
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Variational iteration method – a kind of non-linear analytical technique: some examples

1999-07-01
Ji‐Huan He

The finite element method with weighted basis functions for singularly perturbed convection–diffusion problems

2003-12-04
Xiang‐Gui Li , Chi Kin Chan , Song Wang

The Lie-group shooting method for solving the Bratu equation

2011-04-15
S. Abbasbandy , Mir Sajjad Hashemi , Chein‐Shan Liu
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New smoother to enhance multigrid-based methods for Bratu problem

2008-07-14
A. Mohsen , L.F. Sedeek , S.A. Mohamed

An iterative method for solving nonlinear functional equations

2005-06-10
Varsha Daftardar‐Gejji , Hossein Jafari

How accurate is the streamline-diffusion FEM inside characteristic (boundary and interior) layers?

2004-07-30
Natalia Kopteva
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Bifurcations of travelling wave solutions for the Gilson–Pickering equation

2008-08-04
Aiyong Chen , Wentao Huang , Shengqiang Tang

Application of He’s homotopy perturbation method for solving the Cauchy reaction–diffusion problem

2009-01-07
Ahmet Yıldırım

An Efficient Method to Solve Thermal Wave Equation

2014-01-01
Mohamed Salah , R. M. Amer , M. S. Matbuly
In this paper, an efficient technique of differential quadrature method and perturbation method is employed to analyze reaction-diffusion problems. An efficient method is presented to solve thermal wave propagation model 
 In this paper, an efficient technique of differential quadrature method and perturbation method is employed to analyze reaction-diffusion problems. An efficient method is presented to solve thermal wave propagation model in one and two dimensions. The proposed method marches in the time direction block by block and there are several time levels in each block. The global method of differential quadrature is applied in each block to discretize both the spatial and temporal derivatives. Furthermore, the proposed method is validated by comparing the obtained results with the available analytical ones and also compared with the hybrid technique of differential quadrature method and Runge-Kutta fourth order method.
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An Implicit Factored Scheme for the Compressible Navier-Stokes Equations

1978-04-01
Richard M. Beam , R. F. Warming
An implicit finite-difference scheme is developed for the numerical solution of the compressible Navier-Stokes equations in conservation- law form. The algorithm is second-order- time accurate, noniterative, and spatially factored. In 
 An implicit finite-difference scheme is developed for the numerical solution of the compressible Navier-Stokes equations in conservation- law form. The algorithm is second-order- time accurate, noniterative, and spatially factored. In order to obtain an efficient factored algorithm, the spatial cross derivatives are evaluated explicitly. However, the algorithm is unconditional ly stable and, although a three-time-lev el scheme, requires only two time levels of data storage. The algorithm is constructed in a form (i.e., increments of the conserved variables and fluxes) that provides a direct derivation of the scheme and leads to an efficient computational algorithm. In addition, the delta form has the advantageous property of a steady state (if one exists) independent of the size of the time step. Numerical results are presented for a two-dimensiona l shock boundary-layer interaction problem.

Non-perturbative analytical solutions of the space- and time-fractional Burgers equations

2005-10-21
Shaher Momani

MULTIPLE SOLUTIONS OF A GENERALIZED SINGULAR PERTURBED BRATU PROBLEM

2012-03-14
Taoufik Bakri , Yuri A. Kuznetsov , Ferdinand Verhulst +1 more
Nonlinear two-point boundary value problems (BVPs) may have none or more than one solution. For the singularly perturbed two-point BVP Δu″ + 2uâ€Č + f(u) = 0, 0 < x 
 Nonlinear two-point boundary value problems (BVPs) may have none or more than one solution. For the singularly perturbed two-point BVP Δu″ + 2uâ€Č + f(u) = 0, 0 < x < 1, u(0) = 0, u(1) = 0, a condition is given to have one and only one solution; also cases of more solutions have been analyzed. After attention to the form and validity of the corresponding asymptotic expansions, partially based on slow manifold theory, we reconsider the BVP within the framework of small and large values of the parameter. In the case of a special nonlinearity, numerical bifurcation patterns are studied that improve our understanding of the multivaluedness of the solutions.
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High‐order difference schemes for two‐dimensional elliptic equations

1985-03-01
Murli M. Gupta , Ram P. Manohar , J. W. Stephenson
Abstract A high‐order finite‐difference approximation is proposed for numerical solution of linear or quasilinear elliptic differential equation. The approximation is defined on a square mesh stencil using nine node points 
 Abstract A high‐order finite‐difference approximation is proposed for numerical solution of linear or quasilinear elliptic differential equation. The approximation is defined on a square mesh stencil using nine node points and has a truncation error of order h 4 . Several test problems, including one modeling convection‐dominated flows, are solved using this and existing methods. The results clearly exhibit the superiority of the new approximation, in terms of both accuracy and computational efficiency.

Taylor's Decomposition on two points for one-dimensional Bratu problem

2009-06-22
Meltem ADIYAMAN , ƞ. Somali
In this article, Taylor's Decomposition method is introduced for solving one-dimensional Bratu problem. The numerical scheme is based on the application of the Taylor's decomposition to the corresponding first order 
 In this article, Taylor's Decomposition method is introduced for solving one-dimensional Bratu problem. The numerical scheme is based on the application of the Taylor's decomposition to the corresponding first order differential equation system. The technique is illustrated with different eigenvalues and the results show that the method converges rapidly and hence approximate the exact solution very accurately for relatively large step sizes. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010

Numerical computation of three‐dimensional incompressible Navier–Stokes equations in primitive variable form by DQ method

2003-09-02
C. Shu , Lan Wang , Y. T. Chew
Abstract In this paper, the global method of differential quadrature (DQ) is applied to solve three‐dimensional Navier–Stokes equations in primitive variable form on a non‐staggered grid. Two numerical approaches were 
 Abstract In this paper, the global method of differential quadrature (DQ) is applied to solve three‐dimensional Navier–Stokes equations in primitive variable form on a non‐staggered grid. Two numerical approaches were proposed in this work, which are based on the pressure correction process with DQ discretization. The essence in these approaches is the requirement that the continuity equation must be satisfied on the boundary. Meanwhile, suitable boundary condition for pressure correction equation was recommended. Through a test problem of three‐dimensional driven cavity flow, the performance of two approaches was comparatively studied in terms of the accuracy. The numerical results were obtained for Reynolds numbers of 100, 200, 400 and 1000. The present results were compared well with available data in the literature. In this work, the grid‐dependence study was done, and the benchmark solutions for the velocity profiles along the vertical and horizontal centrelines were given. Copyright © 2003 John Wiley & Sons, Ltd.