Finite element method (FEM) is an efficient numerical tool for the solution of partial differential equations (PDEs). It is one of the most general methods when compared to other numerical techniques. PDEs posed in a ...Finite element method (FEM) is an efficient numerical tool for the solution of partial differential equations (PDEs). It is one of the most general methods when compared to other numerical techniques. PDEs posed in a variational form over a given space, say a Hilbert space, are better numerically handled with the FEM. The FEM algorithm is used in various applications which includes fluid flow, heat transfer, acoustics, structural mechanics and dynamics, electric and magnetic field, etc. Thus, in this paper, the Finite Element Orthogonal Collocation Approach (FEOCA) is established for the approximate solution of Time Fractional Telegraph Equation (TFTE) with Mamadu-Njoseh polynomials as grid points corresponding to new basis functions constructed in the finite element space. The FEOCA is an elegant mixture of the Finite Element Method (FEM) and the Orthogonal Collocation Method (OCM). Two numerical examples are experimented on to verify the accuracy and rate of convergence of the method as compared with the theoretical results, and other methods in literature.展开更多
This paper is devoted to studying the asymptotic theory of initial value problems for a semilinear perturbed telegraph equation. The asymptotic theory and validity of formal approximations are constructed on long time...This paper is devoted to studying the asymptotic theory of initial value problems for a semilinear perturbed telegraph equation. The asymptotic theory and validity of formal approximations are constructed on long timescale O(\epsilon\(-1)). As an application of the asymptotic theory, the initial value problems for a special telegraph equation are studied and two asymptotic solutions of order O(\epsilon\(-)1) are presented.展开更多
In this paper,a proficient numerical technique for the time-fractional telegraph equation(TFTE)is proposed.The chief aim of this paper is to utilize a relatively new type of B-spline called the cubic trigonometric B-s...In this paper,a proficient numerical technique for the time-fractional telegraph equation(TFTE)is proposed.The chief aim of this paper is to utilize a relatively new type of B-spline called the cubic trigonometric B-spline for the proposed scheme.This technique is based on finite difference formulation for the Caputo time-fractional derivative and cubic trigonometric B-splines based technique for the derivatives in space.A stability analysis of the scheme is presented to confirm that the errors do not amplify.A convergence analysis is also presented.Computational experiments are carried out in addition to verify the theoretical analysis.Numerical results are contrasted with a few present techniques and it is concluded that the presented scheme is progressively right and more compelling.展开更多
This work is concerned with the application of a redefined set of extended uniform cubic B-spline(RECBS)functions for the numerical treatment of time-fractional Telegraph equation.The presented technique engages finit...This work is concerned with the application of a redefined set of extended uniform cubic B-spline(RECBS)functions for the numerical treatment of time-fractional Telegraph equation.The presented technique engages finite difference formulation for discretizing the Caputo time-fractional derivatives and RECBS functions to interpolate the solution curve along the spatial grid.Stability analysis of the scheme is provided to ensure that the errors do not amplify during the execution of the numerical procedure.The derivation of uniform convergence has also been presented.Some computational experiments are executed to verify the theoretical considerations.Numerical results are compared with the existing schemes and it is concluded that the present scheme returns superior outcomes on the topic.展开更多
Numerical solutions of the second-order one-dimensional hyperbolic telegraph equations are presented using the radial basis functions.The purpose of this paper is to propose a simple novel direct meshless scheme for s...Numerical solutions of the second-order one-dimensional hyperbolic telegraph equations are presented using the radial basis functions.The purpose of this paper is to propose a simple novel direct meshless scheme for solving hyperbolic telegraph equations.This is fulfilled by considering time variable as normal space variable.Under this scheme,there is no need to remove time-dependent variable during the whole solution process.Since the numerical solution accuracy depends on the condition of coefficient matrix derived from the radial basis function method.We propose a simple shifted domain method,which can avoid the full-coefficient interpolation matrix easily.Numerical experiments performed with the proposed numerical scheme for several second-order hyperbolic telegraph equations are presented with some discussions.展开更多
Telegraph equations are derived from the equations of transmission line theory. They describe the relationships between the currents and voltages on a portion of an electric line as a function of the linear constants ...Telegraph equations are derived from the equations of transmission line theory. They describe the relationships between the currents and voltages on a portion of an electric line as a function of the linear constants of the conductor (resistance, conductance, inductance, capacitance). Their resolution makes it possible to determine the variation of the current and the voltage as a function of time at each point of the line. By adopting a general sinusoidal form, we propose a new exact solution to the telegraphers’ partial differential equations. Different simulations have been carried out considering the parameter of the 12/20 (24) kV Medium Voltage Cable NF C 33,220. The curves of the obtained solution better fit the real voltage curves observed in the electrical networks in operation.展开更多
In this paper, we examine the space discretization of time fractional telegraph equation (TFTE) with Mamadu-Njoseh orthogonal basis functions. For ease and convenience, we deal with the fractional derivative by first ...In this paper, we examine the space discretization of time fractional telegraph equation (TFTE) with Mamadu-Njoseh orthogonal basis functions. For ease and convenience, we deal with the fractional derivative by first converting from Caputo’s type to Riemann-Liouville’s type. The proposed method was constrained to precise error analysis to establish the accuracy of the method. Numerical experimentation was implemented with the aid of MAPLE 18 to show convergence of the method as compared with the analytic solution.展开更多
In this paper, an approximate solution for the one-dimensional hyperbolic telegraph equation by using the q-homotopy analysis method （q-HAM） is proposed.The results shows that the convergence of the q- homotopy anal...In this paper, an approximate solution for the one-dimensional hyperbolic telegraph equation by using the q-homotopy analysis method （q-HAM） is proposed.The results shows that the convergence of the q- homotopy analysis method is more accurate than the convergence of the homotopy analysis method （HAM）.展开更多
Telegraph equations describing the particle densities in Brownian movement on a lattice site have been derived and it has been shown that the complementary classical Dirac equation appears naturally as the consequence...Telegraph equations describing the particle densities in Brownian movement on a lattice site have been derived and it has been shown that the complementary classical Dirac equation appears naturally as the consequence of correlations in particle trajectories in Brownian movement. It has also been demonstrated that Heisenberg uncertainty relation between energy and time is the necessary and sufficient condition to transform this classical equation into usual Dirac’s relativistic quantum equation.展开更多
Over the last few years, there has been a significant increase in attention paid to fractional differential equations, given their wide array of applications in the fields of physics and engineering. The recent develo...Over the last few years, there has been a significant increase in attention paid to fractional differential equations, given their wide array of applications in the fields of physics and engineering. The recent development of using fractional telegraph equations as models in some fields (e.g., the thermal diffusion in fractal media) has heightened the importance of examining the method of solutions for such equations (both approximate and analytic). The present work is designed to serve as a valuable contribution to work in this field. The key objective of this work is to propose a general framework that can be used to guide quadratic spline functions in order to create a numerical method for obtaining an approximation solution using the linear space-fractional telegraph equation. Additionally, the Von Neumann method was employed to measure the stability of the analytical scheme, which showed that the proposed method is conditionally stable. What’s more, the proposal contains a numerical example that illustrates how the proposed method can be implemented practically, whilst the error estimates and numerical stability results are discussed in depth. The findings indicate that the proposed model is highly effective, convenient and accurate for solving the relevant problems and is suitable for use with approximate solutions acquired through the two-dimensional differential transform method that has been developed for linear partial differential equations with space- and time-fractional derivatives.展开更多
The space-fractional telegraph equation is analyzed and the Fourier transform of its funda-mental solution is obtained and discussed.A symmetric process with discontinuous trajectories, whose transition function satis...The space-fractional telegraph equation is analyzed and the Fourier transform of its funda-mental solution is obtained and discussed.A symmetric process with discontinuous trajectories, whose transition function satisfies thespace-fractional telegraph equation, is presented. Its limiting behaviour and the connectionwith symmetric stable processes is also examined.展开更多
A new numerical algorithm for telegraph equations with homogeneous boundary con- ditions is proposed. Due to the damping terms in telegraph equations, there is no royal conservation law according to Noether＇s theorem...A new numerical algorithm for telegraph equations with homogeneous boundary con- ditions is proposed. Due to the damping terms in telegraph equations, there is no royal conservation law according to Noether＇s theorem. The algorithm origins from the discovery of a transform applied to a telegraph equation, which transforms the telegraph equation to a Klein-Gordon equation. The Symplectic method is then brought in this algorithm to solve the Klein-Gordon equation, which is based on the fact that the Klein-Gordon equation with the homogeneous boundary condition is a perfect Hamiltonian system and the symplectic method works very well for Hamiltonian systems. The transformation itself and the inverse transformation theoretically bring no error to the numerical computation. Therefore the error only comes from the symplectic scheme chosen. The telegraph equation is finally explicitly computed when an explicit symplectic scheme is utilized. A relatively long time result can be expected due to the application of the symplectic method. Mean- while, we present order analysis for both one-dimensional and multi-dimensional cases in the paper. The efficiency of this approach is demonstrated with numerical examples.展开更多
In this paper,we study the numerical solution of the time-fractional telegraph equation on the unbounded domain.We first introduce the artificial boundariesГ±to get a finite computational domain.On the artificia...In this paper,we study the numerical solution of the time-fractional telegraph equation on the unbounded domain.We first introduce the artificial boundariesГ±to get a finite computational domain.On the artificial boundariesГ±,we use the Laplace transform to construct the exact artificial boundary conditions(ABCs)to reduce the original problem to an initial-boundary value problem on a bounded domain.In addition,we propose a finite difference scheme based on the L_(1−2)formule for the Caputo fractional derivative in time direction and the central difference scheme for the spatial directional derivative to solve the reduced problem.In order to reduce the effect of unsmoothness of the solution at the initial moment,we use a fine mesh and low-order interpolation to discretize the solution near t=0.Finally,some numerical results show the efficiency and reliability of the ABCs and validate our theoretical results.展开更多
In this paper, we study the semi-boundless mixed problem for time-fractional telegraph equation. We are able to use the integral transform method （the Fourier sin and cos transforms） to obtain the solution.
In this paper,we develop an accurate and efficient Legendre wavelets method for numerical solution of the well known time-fractional telegraph equation.In the proposed method we have employed both of the operational m...In this paper,we develop an accurate and efficient Legendre wavelets method for numerical solution of the well known time-fractional telegraph equation.In the proposed method we have employed both of the operational matrices of fractional integration and differentiation to get numerical solution of the time-telegraph equation.The power of this manageable method is confirmed.Moreover the use of Legendre wavelet is found to be accurate,simple and fast.展开更多
Based on classical circuit theory, this article develops a general analytic solution of the telegrapher’s equations, in which the length of the cable is explicitly contained as a freely adjustable parameter. For this...Based on classical circuit theory, this article develops a general analytic solution of the telegrapher’s equations, in which the length of the cable is explicitly contained as a freely adjustable parameter. For this reason, the solution is also applicable to electrically short cables. Such a model has become indispensable because a few months ago, it was experimentally shown that voltage fluctuations in ordinary but electrically short copper lines move at signal velocities that are significantly higher than the speed of light in a vacuum. This finding contradicts the statements of the special theory of relativity but not, as is shown here, the fundamental principles of electrical engineering. Based on the general transfer function of a transmission line, the article shows mathematically that an unterminated, electrically short cable has the characteristics of an ideal delay element, meaning that an input signal appears at the output with a slight delay but remains otherwise unchanged. Even for conventional cables, the time constants can be so small that the corresponding signal velocities can significantly exceed the speed of light in a vacuum. The article also analyses the technical means with which this effect can be conveyed to very long cables.展开更多
It is known that Fourier’s heat equation, which is parabolic, implies an infinite velocity propagation, or, in other words, that the mechanism of heat conduction is established instantaneously under all conditions. T...It is known that Fourier’s heat equation, which is parabolic, implies an infinite velocity propagation, or, in other words, that the mechanism of heat conduction is established instantaneously under all conditions. This is unacceptable on physical grounds in spite of the fact that Fourier’s law agrees well with experiment. However, discrepancies are likely to occur when extremely short distances or extremely short time intervals are considered, as they must in some modern problems of aero-thermodynamics. Cattaneo and independently Vernotte proved that such process can be described by Heaviside’s telegraph equation. This paper shows that this fact can be derived using calculus of variations, by application of the Euler-Lagrange equation. So, we proved that the equation of heat conduction with finite velocity propagation of the thermal disturbance can be obtained as a solution to one variational problem.展开更多
This work deals with a second order linear general equation with partial derivatives for a two-variable function. It covers a wide range of applications. This equation is solved with a finite difference hybrid method:...This work deals with a second order linear general equation with partial derivatives for a two-variable function. It covers a wide range of applications. This equation is solved with a finite difference hybrid method: BTCS + CTCS. This scheme is simple, precise, and economical in terms of time and space occupancy in memory.展开更多
In this paper,we develop a lattice Boltzmann model for a class ofone-dimensional nonlinear wave equations,including the second-order hyperbolictelegraph equation,the nonlinear Klein-Gordon equation,the damped and unda...In this paper,we develop a lattice Boltzmann model for a class ofone-dimensional nonlinear wave equations,including the second-order hyperbolictelegraph equation,the nonlinear Klein-Gordon equation,the damped and undampedsine-Gordon equation and double sine-Gordon equation.By choosing properly theconservation condition between the macroscopic quantity u,and the distributionfunctions and applying the Chapman-Enskog expansion,the governing equation isrecovered correctly from the lattice Boltzmann equation.Moreover,the local equilib-rium distribution function is obtained.The results of numerical examples have beencompared with the analytical solutions to confirm the good accuracy and the applica-bility of our scheme.展开更多
In this paper,we propose a new high accuracy discretization based on the ideas given by Chawla and Shivakumar for the solution of two-space dimensional nonlinear hyper-bolic partial differential equation of the form u...In this paper,we propose a new high accuracy discretization based on the ideas given by Chawla and Shivakumar for the solution of two-space dimensional nonlinear hyper-bolic partial differential equation of the form utt=A(x,y,t)uxx+B(x,y,t)uyy+g(x,y,t,u,ux,uy,ut),0<x,y<1,t>0 subject to appropriate initial and Dirichlet boundary conditions.We use only five evaluations of the function g and do not require any fictitious points to discretize the differential equation.The proposed method is directly applicable to wave equation in polar coordinates and when applied to a linear telegraphic hyperbolic equation is shown to be unconditionally stable.Numerical results are provided to illustrate the usefulness of the proposed method.展开更多
文摘Finite element method (FEM) is an efficient numerical tool for the solution of partial differential equations (PDEs). It is one of the most general methods when compared to other numerical techniques. PDEs posed in a variational form over a given space, say a Hilbert space, are better numerically handled with the FEM. The FEM algorithm is used in various applications which includes fluid flow, heat transfer, acoustics, structural mechanics and dynamics, electric and magnetic field, etc. Thus, in this paper, the Finite Element Orthogonal Collocation Approach (FEOCA) is established for the approximate solution of Time Fractional Telegraph Equation (TFTE) with Mamadu-Njoseh polynomials as grid points corresponding to new basis functions constructed in the finite element space. The FEOCA is an elegant mixture of the Finite Element Method (FEM) and the Orthogonal Collocation Method (OCM). Two numerical examples are experimented on to verify the accuracy and rate of convergence of the method as compared with the theoretical results, and other methods in literature.
文摘This paper is devoted to studying the asymptotic theory of initial value problems for a semilinear perturbed telegraph equation. The asymptotic theory and validity of formal approximations are constructed on long timescale O(\epsilon\(-1)). As an application of the asymptotic theory, the initial value problems for a special telegraph equation are studied and two asymptotic solutions of order O(\epsilon\(-)1) are presented.
文摘In this paper,a proficient numerical technique for the time-fractional telegraph equation(TFTE)is proposed.The chief aim of this paper is to utilize a relatively new type of B-spline called the cubic trigonometric B-spline for the proposed scheme.This technique is based on finite difference formulation for the Caputo time-fractional derivative and cubic trigonometric B-splines based technique for the derivatives in space.A stability analysis of the scheme is presented to confirm that the errors do not amplify.A convergence analysis is also presented.Computational experiments are carried out in addition to verify the theoretical analysis.Numerical results are contrasted with a few present techniques and it is concluded that the presented scheme is progressively right and more compelling.
文摘This work is concerned with the application of a redefined set of extended uniform cubic B-spline(RECBS)functions for the numerical treatment of time-fractional Telegraph equation.The presented technique engages finite difference formulation for discretizing the Caputo time-fractional derivatives and RECBS functions to interpolate the solution curve along the spatial grid.Stability analysis of the scheme is provided to ensure that the errors do not amplify during the execution of the numerical procedure.The derivation of uniform convergence has also been presented.Some computational experiments are executed to verify the theoretical considerations.Numerical results are compared with the existing schemes and it is concluded that the present scheme returns superior outcomes on the topic.
基金The first author is supported by the Natural Science Foundation of Anhui Province(Project No.1908085QA09)the University Natural Science Research Project of Anhui Province(Project Nos.KJ2019A0591&KJ2020B06)。
文摘Numerical solutions of the second-order one-dimensional hyperbolic telegraph equations are presented using the radial basis functions.The purpose of this paper is to propose a simple novel direct meshless scheme for solving hyperbolic telegraph equations.This is fulfilled by considering time variable as normal space variable.Under this scheme,there is no need to remove time-dependent variable during the whole solution process.Since the numerical solution accuracy depends on the condition of coefficient matrix derived from the radial basis function method.We propose a simple shifted domain method,which can avoid the full-coefficient interpolation matrix easily.Numerical experiments performed with the proposed numerical scheme for several second-order hyperbolic telegraph equations are presented with some discussions.
文摘Telegraph equations are derived from the equations of transmission line theory. They describe the relationships between the currents and voltages on a portion of an electric line as a function of the linear constants of the conductor (resistance, conductance, inductance, capacitance). Their resolution makes it possible to determine the variation of the current and the voltage as a function of time at each point of the line. By adopting a general sinusoidal form, we propose a new exact solution to the telegraphers’ partial differential equations. Different simulations have been carried out considering the parameter of the 12/20 (24) kV Medium Voltage Cable NF C 33,220. The curves of the obtained solution better fit the real voltage curves observed in the electrical networks in operation.
文摘In this paper, we examine the space discretization of time fractional telegraph equation (TFTE) with Mamadu-Njoseh orthogonal basis functions. For ease and convenience, we deal with the fractional derivative by first converting from Caputo’s type to Riemann-Liouville’s type. The proposed method was constrained to precise error analysis to establish the accuracy of the method. Numerical experimentation was implemented with the aid of MAPLE 18 to show convergence of the method as compared with the analytic solution.
文摘In this paper, an approximate solution for the one-dimensional hyperbolic telegraph equation by using the q-homotopy analysis method （q-HAM） is proposed.The results shows that the convergence of the q- homotopy analysis method is more accurate than the convergence of the homotopy analysis method （HAM）.
文摘Telegraph equations describing the particle densities in Brownian movement on a lattice site have been derived and it has been shown that the complementary classical Dirac equation appears naturally as the consequence of correlations in particle trajectories in Brownian movement. It has also been demonstrated that Heisenberg uncertainty relation between energy and time is the necessary and sufficient condition to transform this classical equation into usual Dirac’s relativistic quantum equation.
文摘Over the last few years, there has been a significant increase in attention paid to fractional differential equations, given their wide array of applications in the fields of physics and engineering. The recent development of using fractional telegraph equations as models in some fields (e.g., the thermal diffusion in fractal media) has heightened the importance of examining the method of solutions for such equations (both approximate and analytic). The present work is designed to serve as a valuable contribution to work in this field. The key objective of this work is to propose a general framework that can be used to guide quadratic spline functions in order to create a numerical method for obtaining an approximation solution using the linear space-fractional telegraph equation. Additionally, the Von Neumann method was employed to measure the stability of the analytical scheme, which showed that the proposed method is conditionally stable. What’s more, the proposal contains a numerical example that illustrates how the proposed method can be implemented practically, whilst the error estimates and numerical stability results are discussed in depth. The findings indicate that the proposed model is highly effective, convenient and accurate for solving the relevant problems and is suitable for use with approximate solutions acquired through the two-dimensional differential transform method that has been developed for linear partial differential equations with space- and time-fractional derivatives.
基金Project supported by the National Natural Science Foundation of China (No. 10071014).
文摘The space-fractional telegraph equation is analyzed and the Fourier transform of its funda-mental solution is obtained and discussed.A symmetric process with discontinuous trajectories, whose transition function satisfies thespace-fractional telegraph equation, is presented. Its limiting behaviour and the connectionwith symmetric stable processes is also examined.
基金Acknowledgments. This work is supported by the Natural Science Foundation of China （NSFC） under grant 11371287 and the International Science and Technology Cooperation Program of China under grant 2010DFA14700.
文摘A new numerical algorithm for telegraph equations with homogeneous boundary con- ditions is proposed. Due to the damping terms in telegraph equations, there is no royal conservation law according to Noether＇s theorem. The algorithm origins from the discovery of a transform applied to a telegraph equation, which transforms the telegraph equation to a Klein-Gordon equation. The Symplectic method is then brought in this algorithm to solve the Klein-Gordon equation, which is based on the fact that the Klein-Gordon equation with the homogeneous boundary condition is a perfect Hamiltonian system and the symplectic method works very well for Hamiltonian systems. The transformation itself and the inverse transformation theoretically bring no error to the numerical computation. Therefore the error only comes from the symplectic scheme chosen. The telegraph equation is finally explicitly computed when an explicit symplectic scheme is utilized. A relatively long time result can be expected due to the application of the symplectic method. Mean- while, we present order analysis for both one-dimensional and multi-dimensional cases in the paper. The efficiency of this approach is demonstrated with numerical examples.
基金supported by the NSFC Projects No.12025104,11871298the Scientific Research Foundation of NUAA No.YAH21109.
文摘In this paper,we study the numerical solution of the time-fractional telegraph equation on the unbounded domain.We first introduce the artificial boundariesГ±to get a finite computational domain.On the artificial boundariesГ±,we use the Laplace transform to construct the exact artificial boundary conditions(ABCs)to reduce the original problem to an initial-boundary value problem on a bounded domain.In addition,we propose a finite difference scheme based on the L_(1−2)formule for the Caputo fractional derivative in time direction and the central difference scheme for the spatial directional derivative to solve the reduced problem.In order to reduce the effect of unsmoothness of the solution at the initial moment,we use a fine mesh and low-order interpolation to discretize the solution near t=0.Finally,some numerical results show the efficiency and reliability of the ABCs and validate our theoretical results.
文摘In this paper, we study the semi-boundless mixed problem for time-fractional telegraph equation. We are able to use the integral transform method （the Fourier sin and cos transforms） to obtain the solution.
文摘In this paper,we develop an accurate and efficient Legendre wavelets method for numerical solution of the well known time-fractional telegraph equation.In the proposed method we have employed both of the operational matrices of fractional integration and differentiation to get numerical solution of the time-telegraph equation.The power of this manageable method is confirmed.Moreover the use of Legendre wavelet is found to be accurate,simple and fast.
文摘Based on classical circuit theory, this article develops a general analytic solution of the telegrapher’s equations, in which the length of the cable is explicitly contained as a freely adjustable parameter. For this reason, the solution is also applicable to electrically short cables. Such a model has become indispensable because a few months ago, it was experimentally shown that voltage fluctuations in ordinary but electrically short copper lines move at signal velocities that are significantly higher than the speed of light in a vacuum. This finding contradicts the statements of the special theory of relativity but not, as is shown here, the fundamental principles of electrical engineering. Based on the general transfer function of a transmission line, the article shows mathematically that an unterminated, electrically short cable has the characteristics of an ideal delay element, meaning that an input signal appears at the output with a slight delay but remains otherwise unchanged. Even for conventional cables, the time constants can be so small that the corresponding signal velocities can significantly exceed the speed of light in a vacuum. The article also analyses the technical means with which this effect can be conveyed to very long cables.
文摘It is known that Fourier’s heat equation, which is parabolic, implies an infinite velocity propagation, or, in other words, that the mechanism of heat conduction is established instantaneously under all conditions. This is unacceptable on physical grounds in spite of the fact that Fourier’s law agrees well with experiment. However, discrepancies are likely to occur when extremely short distances or extremely short time intervals are considered, as they must in some modern problems of aero-thermodynamics. Cattaneo and independently Vernotte proved that such process can be described by Heaviside’s telegraph equation. This paper shows that this fact can be derived using calculus of variations, by application of the Euler-Lagrange equation. So, we proved that the equation of heat conduction with finite velocity propagation of the thermal disturbance can be obtained as a solution to one variational problem.
文摘This work deals with a second order linear general equation with partial derivatives for a two-variable function. It covers a wide range of applications. This equation is solved with a finite difference hybrid method: BTCS + CTCS. This scheme is simple, precise, and economical in terms of time and space occupancy in memory.
基金The authors are very thankful to the reviewers for their valuable suggestions toimprove the quality of the paper.This work is supported by National Natural Science Foundation of China(Nos.11101399,11271171,11301234)the Provincial Natural Science Foundation of Jiangxi(Nos.20161ACB20006,20142BCB23009,20151BAB201012).
文摘In this paper,we develop a lattice Boltzmann model for a class ofone-dimensional nonlinear wave equations,including the second-order hyperbolictelegraph equation,the nonlinear Klein-Gordon equation,the damped and undampedsine-Gordon equation and double sine-Gordon equation.By choosing properly theconservation condition between the macroscopic quantity u,and the distributionfunctions and applying the Chapman-Enskog expansion,the governing equation isrecovered correctly from the lattice Boltzmann equation.Moreover,the local equilib-rium distribution function is obtained.The results of numerical examples have beencompared with the analytical solutions to confirm the good accuracy and the applica-bility of our scheme.
基金“The University of Delhi”under research grant No.Dean(R)/R&D/2010/1311.
文摘In this paper,we propose a new high accuracy discretization based on the ideas given by Chawla and Shivakumar for the solution of two-space dimensional nonlinear hyper-bolic partial differential equation of the form utt=A(x,y,t)uxx+B(x,y,t)uyy+g(x,y,t,u,ux,uy,ut),0<x,y<1,t>0 subject to appropriate initial and Dirichlet boundary conditions.We use only five evaluations of the function g and do not require any fictitious points to discretize the differential equation.The proposed method is directly applicable to wave equation in polar coordinates and when applied to a linear telegraphic hyperbolic equation is shown to be unconditionally stable.Numerical results are provided to illustrate the usefulness of the proposed method.
