In recent papers the solution of nonlinear Fredholm integral equations was discussed using Adomian decomposition method (ADM). For case in which the integrals are analytically impossible, ADM can not be applied. In th...In recent papers the solution of nonlinear Fredholm integral equations was discussed using Adomian decomposition method (ADM). For case in which the integrals are analytically impossible, ADM can not be applied. In this paper a discretized version of the ADM is introduced and the proposed version will be called discrete Adomian decomposition method (DADM). An accelerated formula of Adomian polynomials is used in calculations. Based on this formula, a new convergence approach of ADM is introduced. Convergence approach is reliable enough to obtain an explicit formula for the maximum absolute truncated error of the Adomian’s series solution. Also, we prove that the solution of nonlinear Fredholm integral equation by DADM converges to ADM solution. Finally, some numerical examples were introduced.展开更多
The rotating shallow water system is an important physical model,which has been widely used in many scientific areas,such as fluids,hydrodynamics,geophysics,oceanic and atmospheric dynamics.In this paper,we extend the...The rotating shallow water system is an important physical model,which has been widely used in many scientific areas,such as fluids,hydrodynamics,geophysics,oceanic and atmospheric dynamics.In this paper,we extend the application of the Adomian decomposition method from the single equation to the coupled system to investigate the numerical solutions of the rotating shallow water system with an underlying circular paraboloidal basin.By introducing some special initial values,we obtain interesting approximate pulsrodon solutions corresponding to pulsating elliptic warm-core rings,which take the form of realistic series solutions.Numerical results reveal that the numerical pulsrodon solutions can quickly converge to the exact solutions derived by Rogers and An,which fully shows the efficiency and accuracy of the proposed method.Note that the method proposed can be effectively used to construct numerical solutions of many nonlinear mathematical physics equations.The results obtained provide some potential theoretical guidance for experts to study the related phenomena in geography,oceanic and atmospheric science.展开更多
This study presents a numerical investigation of shallow water wave dynamics with particular emphasis on the role of surface tension.In the absence of surface tension,shallow water waves are primarily driven by gravit...This study presents a numerical investigation of shallow water wave dynamics with particular emphasis on the role of surface tension.In the absence of surface tension,shallow water waves are primarily driven by gravity and are well described by the classical Boussinesq equation,which incorporates fourth-order dispersion.Under this framework,solitary and shock waves arise through the balance of nonlinearity and gravity-induced dispersion,producing waveforms whose propagation speed,amplitude,and width depend largely on depth and initial disturbance.The resulting dynamics are comparatively smoother,with solitary waves maintaining coherent structures and shock waves displaying gradual transitions.When surface tension is incorporated,however,the dynamics become significantly richer.Surface tension introduces additional sixth-order dispersive terms into the governing equation,extending the classical model to the sixth-order Boussinesq equation.This higher-order dispersion modifies the balance between nonlinearity and dispersion,leading to sharper solitary wave profiles,altered shock structures,and a stronger sensitivity of wave stability to parametric variations.Surface tension effects also change the scaling laws for wave amplitude and velocity,producing conditions where solitary waves can narrow while maintaining large amplitudes,or where shock fronts steepen more rapidly compared to the tension-free case.These differences highlight how capillary forces,though often neglected in macroscopic wave studies,play a fundamental role in shaping dynamics at smaller scales or in systems with strong fluid–interface interactions.The analysis in this work is carried out using the Laplace-Adomian Decomposition Method(LADM),chosen for its efficiency and accuracy in solving high-order nonlinear partial differential equations.The numerical scheme successfully recovers both solitary and shock wave solutions under the sixth-order model,with error analysis confirming remarkably low numerical deviations.These results underscore the robustness of the method while demonstrating the profound contrast between shallow water wave dynamics without and with surface tension.展开更多
In this study, the effects of magnetic field and nanoparticle on the Jeffery- Hamel flow are studied using a powerful analytical method called the Adomian decomposition method （ADM）. The traditional Navier-Stokes eq...In this study, the effects of magnetic field and nanoparticle on the Jeffery- Hamel flow are studied using a powerful analytical method called the Adomian decomposition method （ADM）. The traditional Navier-Stokes equation of fluid mechanics and Maxwell＇s electromagnetism governing equations are reduced to nonlinear ordinary differential equations to model the problem. The obtained results are well agreed with that of the Runge-Kutta method. The present plots confirm that the method has high accuracy for different a, Ha, and Re numbers. The flow field inside the divergent channel is studied for various values of Hartmann ：number and angle of channel. The effect of nanoparticle volume fraction in the absence of magnetic field is investigated.展开更多
The aim of this paper is to apply the relatively new Adomian decomposition method to solving the system of linear fractional, in the sense of Riemann-Liouville and Caputo respectively, differential equations. The solu...The aim of this paper is to apply the relatively new Adomian decomposition method to solving the system of linear fractional, in the sense of Riemann-Liouville and Caputo respectively, differential equations. The solutions are expressed in terms of Mittag-Leffier functions of matric argument. The Adomian decomposition method is straightforward, applicable for broader problems and avoids the difficulties in applying integral transforms. As the order is 1, the result here is simplified to that of first order differential equation.展开更多
The Adomian decomposition method （ADM） and Pade approximants are combined to solve the well-known Blaszak-Marciniak lattice, which has rich mathematical structures and many important applications in physics and math...The Adomian decomposition method （ADM） and Pade approximants are combined to solve the well-known Blaszak-Marciniak lattice, which has rich mathematical structures and many important applications in physics and mathematics. In some cases, the truncated series solution of ADM is adequate only in a small region when the exact solution is not reached. To overcome the drawback, the Pade approximants, which have the advantage in turning the polynomials approximation into a rational function, are applied to the series solution to improve the accuracy and enlarge the convergence domain. By using the ADM-Pade technique, the soliton solutions of the Blaszak-Marciniak lattice are constructed with better accuracy and better convergence than by using the ADM alone. Numerical and figurative illustrations show that it is a promising tool for solving nonlinear problems.展开更多
The study reveals analytically on the 3-dimensional viscous time-dependent gyrotactic bioconvection in swirling nanofluid flow past from a rotating disk.It is known that the deformation of the disk is along the radial...The study reveals analytically on the 3-dimensional viscous time-dependent gyrotactic bioconvection in swirling nanofluid flow past from a rotating disk.It is known that the deformation of the disk is along the radial direction.In addition to that Stefan blowing is considered.The Buongiorno nanofluid model is taken care of assuming the fluid to be dilute and we find Brownian motion and thermophoresis have dominant role on nanoscale unit.The primitive mass conservation equation,radial,tangential and axial momentum,heat,nano-particle concentration and micro-organism density function are developed in a cylindrical polar coordinate system with appropriate wall(disk surface)and free stream boundary conditions.This highly nonlinear,strongly coupled system of unsteady partial differential equations is normalized with the classical von Kármán and other transformations to render the boundary value problem into an ordinary differential system.The emerging 11th order system features an extensive range of dimensionless flow parameters,i.e.,disk stretching rate,Brownian motion,thermophoresis,bioconvection Lewis number,unsteadiness parameter,ordinary Lewis number,Prandtl number,mass convective Biot number,Péclet number and Stefan blowing parameter.Solutions of the system are obtained with developed semi-analytical technique,i.e.,Adomian decomposition method.Validation of the said problem is also conducted with earlier literature computed by Runge-Kutta shooting technique.展开更多
In this paper, Adomian decomposition method (ADM) is used to solve the Volterra-Fredholm integral equation. A number of examples have been presented to explain the numerical results, which is the comparison between th...In this paper, Adomian decomposition method (ADM) is used to solve the Volterra-Fredholm integral equation. A number of examples have been presented to explain the numerical results, which is the comparison between the exact solution and the numerical solution, and it is found through the tables and the amount of error between the exact solution and the numerical solution, it is very small and almost non-existent and is also illustrated through the graph how the exact solution of completely applies to the numerical solution This proves the accuracy of the method, which is the Adomian decomposition method (ADM) for solving the Volterra Fredholm integral equation using Maple 18. And that this method is characterized by ease, speed and great accuracy in obtaining numerical results.展开更多
The analytical solution of a viscoelastic continuous beam whose damping characteristics are described in terms of a fractional derivative of arbitrary order was derived by means of the AdoInian decomposition method. T...The analytical solution of a viscoelastic continuous beam whose damping characteristics are described in terms of a fractional derivative of arbitrary order was derived by means of the AdoInian decomposition method. The solution contains arbitrary initial conditions and zero input. For specific analysis, the initial conditions were assumed homogeneous, and the input force was treated as a special process with a particular beam. Two simple cases, step and impulse function responses, were considered respectively. Subsequently, some figures were plotted to show the displacement of the beam under different sets of parameters including different orders of the fractional derivatives.展开更多
This paper presents a method to solve the problems of solutions for integer differential and partial differential equations using the convergence of Adomian's Method. In this paper, we firstly use the convergence ...This paper presents a method to solve the problems of solutions for integer differential and partial differential equations using the convergence of Adomian's Method. In this paper, we firstly use the convergence of Adomian's Method to derive the solutions of high order linear fractional equations, and then the numerical solutions for nonlinear fractional equations. we also get the solutions of two fractional reaction-diffusion equations.We can see the advantage of this method to deal with fractional differential equations.展开更多
The present paper is devoted to the convergence control and accelerating the traditional Decomposition Methodof Adomian (ADM). By means of perturbing the initial or early terms of the Adomian iterates by adding aparam...The present paper is devoted to the convergence control and accelerating the traditional Decomposition Methodof Adomian (ADM). By means of perturbing the initial or early terms of the Adomian iterates by adding aparameterized term, containing an embedded parameter, new modified ADM is constructed. The optimal value ofthis parameter is later determined via squared residual minimizing the error. The failure of the classical ADM is alsoprevented by a suitable value of the embedded parameter, particularly beneficial for the Duan–Rach modification ofthe ADM incorporating all the boundaries into the formulation. With the presented squared residual error analysis,there is no need to check out the results against the numerical ones, as usually has to be done in the traditional ADMstudies to convince the readers that the results are indeed converged to the realistic solutions. Physical examplesselected from the recent application of ADM demonstrate the validity, accuracy and power of the presented novelapproach in this paper. Hence, the highly nonlinear equations arising from engineering applications can be safelytreated by the outlined method for which the classical ADM may fail or be slow to converge.展开更多
We use the modified Adomian decomposition method（ADM） for solving the nonlinear fractional boundary value problem {D α0＋u（x）=f（x,u（x）） ,0〈x〈1,3〈α≤4u（0）=α0, u″（0）=α2u（1）=β0,u″（1）β2where Dα...We use the modified Adomian decomposition method（ADM） for solving the nonlinear fractional boundary value problem {D α0＋u（x）=f（x,u（x）） ,0〈x〈1,3〈α≤4u（0）=α0, u″（0）=α2u（1）=β0,u″（1）β2where Dα 0 ＋u is Caputo fractional derivative and α0, α2, β0, β2 is not zero at all, and f ： [0, 1] × R→R is continuous. The calculated numerical results show reliability and efficiency of the algorithm given. The numerical procedure is tested on lineax and nonlinear problems.展开更多
Combining Adomian decomposition method （ADM） with Pade approximants, we solve two differentiaidifference equations （DDEs）： the relativistic Toda lattice equation and the modified Volterra lattice equation. With t...Combining Adomian decomposition method （ADM） with Pade approximants, we solve two differentiaidifference equations （DDEs）： the relativistic Toda lattice equation and the modified Volterra lattice equation. With the help of symbolic computation Maple, the results obtained by ADM-Pade technique are compared with those obtained by using ADM alone. The numerical results demonstrate that ADM-Pade technique give the approximate solution with faster convergence rate and higher accuracy and relative in larger domain of convergence than using ADM.展开更多
In this paper, exact and numerical solutions are calculated for discrete complex Ginzburg-Landau equation with initial condition by considering the modified Adomian decomposition method （mADM）, which is an efficient...In this paper, exact and numerical solutions are calculated for discrete complex Ginzburg-Landau equation with initial condition by considering the modified Adomian decomposition method （mADM）, which is an efficient method and does not need linearization, weak nonlinearity assumptions or perturbation theory. The numerical solutions are also compared with their corresponding analytical solutions. It is shown that a very good approximation is achieved with the analytical solutions. Finally, the modulational instability is investigated and the corresponding condition is given.展开更多
In this paper, the Combined Laplace Transform-Adomian Decomposition Method is used to solve nth-order integro-differential equations. The results show that the method is very simple and effective.
The nonlinear partial differential equation is solved using the Adomian decomposition method (ADM) in this article. A number of examples have been provided to illustrate the numerical results, which is the comparison ...The nonlinear partial differential equation is solved using the Adomian decomposition method (ADM) in this article. A number of examples have been provided to illustrate the numerical results, which is the comparison of the exact and numerical solutions, and it has been discovered through the tables that the amount of error between the exact and numerical solutions is very small and almost non-existent, and the graph also shows how the exact solution of absolutely applies to the numerical solution. This demonstrates the precision of the Adomian decomposition method (ADM) for solving the nonlinear partial differential equation with Maple18. And that in terms of obtaining numerical results, this approach is characterized by ease, speed, and high accuracy.展开更多
We study approximate solutions of a nonlinear integral equation of Hammerstein type. We describe the principle of discrete Adomian decomposition method (DADM). DADM is considered in the case we evaluate numerical inte...We study approximate solutions of a nonlinear integral equation of Hammerstein type. We describe the principle of discrete Adomian decomposition method (DADM). DADM is considered in the case we evaluate numerical integration by using Chebyshev roots. This technique gives an accurate solutions as will shown by illustrate examples.展开更多
In this paper, we propose a new variation of the Adomian polynomials, which we call the degenerate Adomian polynomials, for the power series solutions of nonlinear ordinary differential equations with nonseparable non...In this paper, we propose a new variation of the Adomian polynomials, which we call the degenerate Adomian polynomials, for the power series solutions of nonlinear ordinary differential equations with nonseparable nonlinearities. We establish efficient algorithms for the degenerate Adomian polynomials. Next we compare the results by the Adomian decomposition method using the classic Adomian polynomials with the results by the Rach-Adomian-Meyers modified decomposition method incorporating the degenerate Adomian polynomials, which itself has been shown to be a confluence of the Adomian decomposition method and the power series method. Convergence acceleration techniques including the diagonal Pade approximants are considered, and new numeric algorithms for the multistage decomposition are deduced using the degenerate Adomian polynomials. Our new technique provides a significant advantage for automated calculations when computing the power series form of the solution for nonlinear ordinary differential equations. Several expository examples are investigated to demonstrate its reliability and efficiency.展开更多
A mathematical model is elaborated for a thermoelastic infinite body with a spherical cavity.A generalized set of governing equations is formulated in the context of three different models of thermoelasticity:the Biot...A mathematical model is elaborated for a thermoelastic infinite body with a spherical cavity.A generalized set of governing equations is formulated in the context of three different models of thermoelasticity:the Biot model,also known as“coupled thermoelasticity”model;the Lord-Shulman model,also referred to as“generalized thermoelasticity with one-relaxation time”approach;and the Green-Lindsay model,also called“generalized thermoelasticity with two-relaxation times”approach.The Adomian’s decomposition method is used to solve the related mathematical problem.The bounding plane of the cavity is subjected to harmonic thermal loading with zero heat flux and strain.Numerical results for the temperature,radial stress,strain,and displacement are represented graphically.It is shown that the angular thermal load and the relaxation times have significant effects on all the studied fields.展开更多
Fractional differential equations are widely used in many fields. In this paper, we discussed the fractional differential equation and the applications of Adomian decomposition method. Where the fractional operator is...Fractional differential equations are widely used in many fields. In this paper, we discussed the fractional differential equation and the applications of Adomian decomposition method. Where the fractional operator is in Caputo sense. Through the numerical test, we can find that the Adomian decomposition method is a powerful tool for solving linear and nonlinear fractional differential equations. The numerical results also show the efficiency of this method.展开更多
文摘In recent papers the solution of nonlinear Fredholm integral equations was discussed using Adomian decomposition method (ADM). For case in which the integrals are analytically impossible, ADM can not be applied. In this paper a discretized version of the ADM is introduced and the proposed version will be called discrete Adomian decomposition method (DADM). An accelerated formula of Adomian polynomials is used in calculations. Based on this formula, a new convergence approach of ADM is introduced. Convergence approach is reliable enough to obtain an explicit formula for the maximum absolute truncated error of the Adomian’s series solution. Also, we prove that the solution of nonlinear Fredholm integral equation by DADM converges to ADM solution. Finally, some numerical examples were introduced.
基金supported by the National Natural Science Foundation of China(Grant No.12371250)the Jiangsu Provincial Natural Science Foundation(Grant Nos.BK20221508,12205154 and 11775116)the Dean’s Research Fund of the Education University of Hong Kong 2023/23(FLASS/DRF/IRS-8)。
文摘The rotating shallow water system is an important physical model,which has been widely used in many scientific areas,such as fluids,hydrodynamics,geophysics,oceanic and atmospheric dynamics.In this paper,we extend the application of the Adomian decomposition method from the single equation to the coupled system to investigate the numerical solutions of the rotating shallow water system with an underlying circular paraboloidal basin.By introducing some special initial values,we obtain interesting approximate pulsrodon solutions corresponding to pulsating elliptic warm-core rings,which take the form of realistic series solutions.Numerical results reveal that the numerical pulsrodon solutions can quickly converge to the exact solutions derived by Rogers and An,which fully shows the efficiency and accuracy of the proposed method.Note that the method proposed can be effectively used to construct numerical solutions of many nonlinear mathematical physics equations.The results obtained provide some potential theoretical guidance for experts to study the related phenomena in geography,oceanic and atmospheric science.
文摘This study presents a numerical investigation of shallow water wave dynamics with particular emphasis on the role of surface tension.In the absence of surface tension,shallow water waves are primarily driven by gravity and are well described by the classical Boussinesq equation,which incorporates fourth-order dispersion.Under this framework,solitary and shock waves arise through the balance of nonlinearity and gravity-induced dispersion,producing waveforms whose propagation speed,amplitude,and width depend largely on depth and initial disturbance.The resulting dynamics are comparatively smoother,with solitary waves maintaining coherent structures and shock waves displaying gradual transitions.When surface tension is incorporated,however,the dynamics become significantly richer.Surface tension introduces additional sixth-order dispersive terms into the governing equation,extending the classical model to the sixth-order Boussinesq equation.This higher-order dispersion modifies the balance between nonlinearity and dispersion,leading to sharper solitary wave profiles,altered shock structures,and a stronger sensitivity of wave stability to parametric variations.Surface tension effects also change the scaling laws for wave amplitude and velocity,producing conditions where solitary waves can narrow while maintaining large amplitudes,or where shock fronts steepen more rapidly compared to the tension-free case.These differences highlight how capillary forces,though often neglected in macroscopic wave studies,play a fundamental role in shaping dynamics at smaller scales or in systems with strong fluid–interface interactions.The analysis in this work is carried out using the Laplace-Adomian Decomposition Method(LADM),chosen for its efficiency and accuracy in solving high-order nonlinear partial differential equations.The numerical scheme successfully recovers both solitary and shock wave solutions under the sixth-order model,with error analysis confirming remarkably low numerical deviations.These results underscore the robustness of the method while demonstrating the profound contrast between shallow water wave dynamics without and with surface tension.
文摘In this study, the effects of magnetic field and nanoparticle on the Jeffery- Hamel flow are studied using a powerful analytical method called the Adomian decomposition method （ADM）. The traditional Navier-Stokes equation of fluid mechanics and Maxwell＇s electromagnetism governing equations are reduced to nonlinear ordinary differential equations to model the problem. The obtained results are well agreed with that of the Runge-Kutta method. The present plots confirm that the method has high accuracy for different a, Ha, and Re numbers. The flow field inside the divergent channel is studied for various values of Hartmann ：number and angle of channel. The effect of nanoparticle volume fraction in the absence of magnetic field is investigated.
基金Supported by the NNSF of China(1027206710461005) the Scientific Research Foundation of Tianjin Education Committee(20050404).
文摘The aim of this paper is to apply the relatively new Adomian decomposition method to solving the system of linear fractional, in the sense of Riemann-Liouville and Caputo respectively, differential equations. The solutions are expressed in terms of Mittag-Leffier functions of matric argument. The Adomian decomposition method is straightforward, applicable for broader problems and avoids the difficulties in applying integral transforms. As the order is 1, the result here is simplified to that of first order differential equation.
基金Project supported by the National Key Basic Research Project of China (Grant No 2004CB318000)the National Natural Science Foundation of China (Grant Nos 10771072 and 10735030)Shanghai Leading Academic Discipline Project of China (Grant No B412)
文摘The Adomian decomposition method （ADM） and Pade approximants are combined to solve the well-known Blaszak-Marciniak lattice, which has rich mathematical structures and many important applications in physics and mathematics. In some cases, the truncated series solution of ADM is adequate only in a small region when the exact solution is not reached. To overcome the drawback, the Pade approximants, which have the advantage in turning the polynomials approximation into a rational function, are applied to the series solution to improve the accuracy and enlarge the convergence domain. By using the ADM-Pade technique, the soliton solutions of the Blaszak-Marciniak lattice are constructed with better accuracy and better convergence than by using the ADM alone. Numerical and figurative illustrations show that it is a promising tool for solving nonlinear problems.
文摘The study reveals analytically on the 3-dimensional viscous time-dependent gyrotactic bioconvection in swirling nanofluid flow past from a rotating disk.It is known that the deformation of the disk is along the radial direction.In addition to that Stefan blowing is considered.The Buongiorno nanofluid model is taken care of assuming the fluid to be dilute and we find Brownian motion and thermophoresis have dominant role on nanoscale unit.The primitive mass conservation equation,radial,tangential and axial momentum,heat,nano-particle concentration and micro-organism density function are developed in a cylindrical polar coordinate system with appropriate wall(disk surface)and free stream boundary conditions.This highly nonlinear,strongly coupled system of unsteady partial differential equations is normalized with the classical von Kármán and other transformations to render the boundary value problem into an ordinary differential system.The emerging 11th order system features an extensive range of dimensionless flow parameters,i.e.,disk stretching rate,Brownian motion,thermophoresis,bioconvection Lewis number,unsteadiness parameter,ordinary Lewis number,Prandtl number,mass convective Biot number,Péclet number and Stefan blowing parameter.Solutions of the system are obtained with developed semi-analytical technique,i.e.,Adomian decomposition method.Validation of the said problem is also conducted with earlier literature computed by Runge-Kutta shooting technique.
文摘In this paper, Adomian decomposition method (ADM) is used to solve the Volterra-Fredholm integral equation. A number of examples have been presented to explain the numerical results, which is the comparison between the exact solution and the numerical solution, and it is found through the tables and the amount of error between the exact solution and the numerical solution, it is very small and almost non-existent and is also illustrated through the graph how the exact solution of completely applies to the numerical solution This proves the accuracy of the method, which is the Adomian decomposition method (ADM) for solving the Volterra Fredholm integral equation using Maple 18. And that this method is characterized by ease, speed and great accuracy in obtaining numerical results.
基金Project supported by the National Natural Science Foundation of China(Nos.10547124 and 10475055)
文摘The analytical solution of a viscoelastic continuous beam whose damping characteristics are described in terms of a fractional derivative of arbitrary order was derived by means of the AdoInian decomposition method. The solution contains arbitrary initial conditions and zero input. For specific analysis, the initial conditions were assumed homogeneous, and the input force was treated as a special process with a particular beam. Two simple cases, step and impulse function responses, were considered respectively. Subsequently, some figures were plotted to show the displacement of the beam under different sets of parameters including different orders of the fractional derivatives.
基金the National Natural Science Foundation of China(Nos.11601003,11371027)Natural Science Research Project of Colleges of Anhui Province(No.KJ2016A023)+1 种基金Natural Science Foundation of Anhui Province(No.1508085MA01)College Students’Scientific Research Training Plan of Anhui University(No.KYXL2014006)
文摘This paper presents a method to solve the problems of solutions for integer differential and partial differential equations using the convergence of Adomian's Method. In this paper, we firstly use the convergence of Adomian's Method to derive the solutions of high order linear fractional equations, and then the numerical solutions for nonlinear fractional equations. we also get the solutions of two fractional reaction-diffusion equations.We can see the advantage of this method to deal with fractional differential equations.
文摘The present paper is devoted to the convergence control and accelerating the traditional Decomposition Methodof Adomian (ADM). By means of perturbing the initial or early terms of the Adomian iterates by adding aparameterized term, containing an embedded parameter, new modified ADM is constructed. The optimal value ofthis parameter is later determined via squared residual minimizing the error. The failure of the classical ADM is alsoprevented by a suitable value of the embedded parameter, particularly beneficial for the Duan–Rach modification ofthe ADM incorporating all the boundaries into the formulation. With the presented squared residual error analysis,there is no need to check out the results against the numerical ones, as usually has to be done in the traditional ADMstudies to convince the readers that the results are indeed converged to the realistic solutions. Physical examplesselected from the recent application of ADM demonstrate the validity, accuracy and power of the presented novelapproach in this paper. Hence, the highly nonlinear equations arising from engineering applications can be safelytreated by the outlined method for which the classical ADM may fail or be slow to converge.
文摘We use the modified Adomian decomposition method（ADM） for solving the nonlinear fractional boundary value problem {D α0＋u（x）=f（x,u（x）） ,0〈x〈1,3〈α≤4u（0）=α0, u″（0）=α2u（1）=β0,u″（1）β2where Dα 0 ＋u is Caputo fractional derivative and α0, α2, β0, β2 is not zero at all, and f ： [0, 1] × R→R is continuous. The calculated numerical results show reliability and efficiency of the algorithm given. The numerical procedure is tested on lineax and nonlinear problems.
基金supported by the National Natural Science Foundation of China under Grant No. 10735030Shanghai Leading Academic Discipline Project under Grant No. B412Program for Changjiang Scholars and Innovative Research Team in University under Grant No. IRT0734
文摘Combining Adomian decomposition method （ADM） with Pade approximants, we solve two differentiaidifference equations （DDEs）： the relativistic Toda lattice equation and the modified Volterra lattice equation. With the help of symbolic computation Maple, the results obtained by ADM-Pade technique are compared with those obtained by using ADM alone. The numerical results demonstrate that ADM-Pade technique give the approximate solution with faster convergence rate and higher accuracy and relative in larger domain of convergence than using ADM.
基金supported by National Natural Science Foundation of China under Grant No. 10672147
文摘In this paper, exact and numerical solutions are calculated for discrete complex Ginzburg-Landau equation with initial condition by considering the modified Adomian decomposition method （mADM）, which is an efficient method and does not need linearization, weak nonlinearity assumptions or perturbation theory. The numerical solutions are also compared with their corresponding analytical solutions. It is shown that a very good approximation is achieved with the analytical solutions. Finally, the modulational instability is investigated and the corresponding condition is given.
文摘In this paper, the Combined Laplace Transform-Adomian Decomposition Method is used to solve nth-order integro-differential equations. The results show that the method is very simple and effective.
文摘The nonlinear partial differential equation is solved using the Adomian decomposition method (ADM) in this article. A number of examples have been provided to illustrate the numerical results, which is the comparison of the exact and numerical solutions, and it has been discovered through the tables that the amount of error between the exact and numerical solutions is very small and almost non-existent, and the graph also shows how the exact solution of absolutely applies to the numerical solution. This demonstrates the precision of the Adomian decomposition method (ADM) for solving the nonlinear partial differential equation with Maple18. And that in terms of obtaining numerical results, this approach is characterized by ease, speed, and high accuracy.
文摘We study approximate solutions of a nonlinear integral equation of Hammerstein type. We describe the principle of discrete Adomian decomposition method (DADM). DADM is considered in the case we evaluate numerical integration by using Chebyshev roots. This technique gives an accurate solutions as will shown by illustrate examples.
文摘In this paper, we propose a new variation of the Adomian polynomials, which we call the degenerate Adomian polynomials, for the power series solutions of nonlinear ordinary differential equations with nonseparable nonlinearities. We establish efficient algorithms for the degenerate Adomian polynomials. Next we compare the results by the Adomian decomposition method using the classic Adomian polynomials with the results by the Rach-Adomian-Meyers modified decomposition method incorporating the degenerate Adomian polynomials, which itself has been shown to be a confluence of the Adomian decomposition method and the power series method. Convergence acceleration techniques including the diagonal Pade approximants are considered, and new numeric algorithms for the multistage decomposition are deduced using the degenerate Adomian polynomials. Our new technique provides a significant advantage for automated calculations when computing the power series form of the solution for nonlinear ordinary differential equations. Several expository examples are investigated to demonstrate its reliability and efficiency.
文摘A mathematical model is elaborated for a thermoelastic infinite body with a spherical cavity.A generalized set of governing equations is formulated in the context of three different models of thermoelasticity:the Biot model,also known as“coupled thermoelasticity”model;the Lord-Shulman model,also referred to as“generalized thermoelasticity with one-relaxation time”approach;and the Green-Lindsay model,also called“generalized thermoelasticity with two-relaxation times”approach.The Adomian’s decomposition method is used to solve the related mathematical problem.The bounding plane of the cavity is subjected to harmonic thermal loading with zero heat flux and strain.Numerical results for the temperature,radial stress,strain,and displacement are represented graphically.It is shown that the angular thermal load and the relaxation times have significant effects on all the studied fields.
文摘Fractional differential equations are widely used in many fields. In this paper, we discussed the fractional differential equation and the applications of Adomian decomposition method. Where the fractional operator is in Caputo sense. Through the numerical test, we can find that the Adomian decomposition method is a powerful tool for solving linear and nonlinear fractional differential equations. The numerical results also show the efficiency of this method.
