Fractional-order time-delay differential equations can describe many complex physical phenomena with memory or delay effects, which are widely used in the fields of cell biology, control systems, signal processing, et...Fractional-order time-delay differential equations can describe many complex physical phenomena with memory or delay effects, which are widely used in the fields of cell biology, control systems, signal processing, etc. Therefore, it is of great significance to study fractional-order time-delay differential equations. In this paper, we discuss a finite volume element method for a class of fractional-order neutral time-delay differential equations. By introducing an intermediate variable, the fourth-order problem is transformed into a system of equations consisting of two second-order partial differential equations. The L1 formula is used to approximate the time fractional order derivative terms, and the finite volume element method is used in space. A fully discrete format of the equations is established, and we prove the existence, uniqueness, convergence and stability of the solution. Finally, the validity of the format is verified by numerical examples.展开更多
In this article, a finite volume element algorithm is presented and discussed for the numerical solutions of a time-fractional nonlinear fourth-order diffusion equation with time delay. By choosing the second-order sp...In this article, a finite volume element algorithm is presented and discussed for the numerical solutions of a time-fractional nonlinear fourth-order diffusion equation with time delay. By choosing the second-order spatial derivative of the original unknown as an additional variable, the fourth-order problem is transformed into a second-order system. Then the fully discrete finite volume element scheme is formulated by using L1approximation for temporal Caputo derivative and finite volume element method in spatial direction. The unique solvability and stable result of the proposed scheme are proved. A priori estimate of L2-norm with optimal order of convergence O(h2+τ2−α)where τand hare time step length and space mesh parameter, respectively, is obtained. The efficiency of the scheme is supported by some numerical experiments.展开更多
Anomalous diffusion is a widespread physical phenomenon,and numerical methods of fractional diffusion models are of important scientific significance and engineering application value.For time fractional diffusion-wav...Anomalous diffusion is a widespread physical phenomenon,and numerical methods of fractional diffusion models are of important scientific significance and engineering application value.For time fractional diffusion-wave equation with damping,a difference(ASC-N,alternating segment Crank-Nicolson)scheme with intrinsic parallelism is proposed.Based on alternating technology,the ASC-N scheme is constructed with four kinds of Saul’yev asymmetric schemes and Crank-Nicolson(C-N)scheme.The unconditional stability and convergence are rigorously analyzed.The theoretical analysis and numerical experiments show that the ASC-N scheme is effective for solving time fractional diffusion-wave equation.展开更多
In this paper, high-order numerical analysis of finite element method（FEM） is presented for twodimensional multi-term time-fractional diffusion-wave equation（TFDWE）. First of all, a fully-discrete approximate sche...In this paper, high-order numerical analysis of finite element method（FEM） is presented for twodimensional multi-term time-fractional diffusion-wave equation（TFDWE）. First of all, a fully-discrete approximate scheme for multi-term TFDWE is established, which is based on bilinear FEM in spatial direction and Crank-Nicolson approximation in temporal direction, respectively. Then the proposed scheme is proved to be unconditionally stable and convergent. And then, rigorous proofs are given here for superclose properties in H-1-norm and temporal convergence in L-2-norm with order O（h-2＋ τ-（3-α））, where h and τ are the spatial size and time step, respectively. At the same time, theoretical analysis of global superconvergence in H-1-norm is derived by interpolation postprocessing technique. At last, numerical example is provided to demonstrate the theoretical analysis.展开更多
In this paper,we study a new numerical technique for a class of 2D nonlinear fractional diffusion-wave equations with the Caputo-type temporal derivative and Riesz-type spatial derivative.Galerkin finite element schem...In this paper,we study a new numerical technique for a class of 2D nonlinear fractional diffusion-wave equations with the Caputo-type temporal derivative and Riesz-type spatial derivative.Galerkin finite element scheme is used for the discretization in the spatial direction,and the temporal component is discretized by a new alternating direction implicit(ADI)method.Next,we strictly prove that the numerical method is stable and convergent.Finally,to confirm our theoretical analysis,some numerical examples in 2D space are presented.展开更多
This article describes the solution procedure of the fractional Pade-Ⅱ equation and generalized Zakharov equation(GSEs)using the sine-cosine method.Pade-Ⅱ is an important nonlinear wave equation modeling unidirectio...This article describes the solution procedure of the fractional Pade-Ⅱ equation and generalized Zakharov equation(GSEs)using the sine-cosine method.Pade-Ⅱ is an important nonlinear wave equation modeling unidirectional propagation of long-wave in dispersive media and GSEs are used to model the interaction between one-dimensional high,and low-frequency waves.Classes of trigonometric and hyperbolic function solutions in fractional calculus are discussed.Graphical simulations of the numerical solutions are flaunted by MATLAB.展开更多
A significant obstacle impeding the advancement of the time fractional Schrodinger equation lies in the challenge of determining its precise mathematical formulation.In order to address this,we undertake an exploratio...A significant obstacle impeding the advancement of the time fractional Schrodinger equation lies in the challenge of determining its precise mathematical formulation.In order to address this,we undertake an exploration of the time fractional Schrodinger equation within the context of a non-Markovian environment.By leveraging a two-level atom as an illustrative case,we find that the choice to raise i to the order of the time derivative is inappropriate.In contrast to the conventional approach used to depict the dynamic evolution of quantum states in a non-Markovian environment,the time fractional Schrodinger equation,when devoid of fractional-order operations on the imaginary unit i,emerges as a more intuitively comprehensible framework in physics and offers greater simplicity in computational aspects.Meanwhile,we also prove that it is meaningless to study the memory of time fractional Schrodinger equation with time derivative 1<α≤2.It should be noted that we have not yet constructed an open system that can be fully described by the time fractional Schrodinger equation.This will be the focus of future research.Our study might provide a new perspective on the role of time fractional Schrodinger equation.展开更多
In this article,we consider the diffusion equation with multi-term time-fractional derivatives.We first derive,by a subordination principle for the solution,that the solution is positive when the initial value is non-...In this article,we consider the diffusion equation with multi-term time-fractional derivatives.We first derive,by a subordination principle for the solution,that the solution is positive when the initial value is non-negative.As an application,we prove the uniqueness of solution to an inverse problem of determination of the temporally varying source term by integral type information in a subdomain.Finally,several numerical experiments are presented to show the accuracy and efficiency of the algorithm.展开更多
Anovel accuratemethod is proposed to solve a broad variety of linear and nonlinear(1+1)-dimensional and(2+1)-dimensional multi-term time-fractional partial differential equations with spatial operators of anisotropic ...Anovel accuratemethod is proposed to solve a broad variety of linear and nonlinear(1+1)-dimensional and(2+1)-dimensional multi-term time-fractional partial differential equations with spatial operators of anisotropic diffusivity.For(1+1)-dimensional problems,analytical solutions that satisfy the boundary requirements are derived.Such solutions are numerically calculated using the trigonometric basis approximation for(2+1)-dimensional problems.With the aid of these analytical or numerical approximations,the original problems can be converted into the fractional ordinary differential equations,and solutions to the fractional ordinary differential equations are approximated by modified radial basis functions with time-dependent coefficients.An efficient backward substitution strategy that was previously provided for a single fractional ordinary differential equation is then used to solve the corresponding systems.The straightforward quasilinearization technique is applied to handle nonlinear issues.Numerical experiments demonstrate the suggested algorithm’s superior accuracy and efficiency.展开更多
This study introduces a pre-orthogonal adaptive Fourier decomposition(POAFD)to obtain approximations and numerical solutions to the fractional Laplacian initial value problem and the extension problem of Caffarelli an...This study introduces a pre-orthogonal adaptive Fourier decomposition(POAFD)to obtain approximations and numerical solutions to the fractional Laplacian initial value problem and the extension problem of Caffarelli and Silvestre(generalized Poisson equation).As a first step,the method expands the initial data function into a sparse series of the fundamental solutions with fast convergence,and,as a second step,makes use of the semigroup or the reproducing kernel property of each of the expanding entries.Experiments show the effectiveness and efficiency of the proposed series solutions.展开更多
In this study,a numerical method based on the Pell-Lucas polynomials(PLPs)is developed to solve the fractional order HIV/AIDS epidemic model with a treatment compartment.The HIV/AIDS mathematical model with a treatmen...In this study,a numerical method based on the Pell-Lucas polynomials(PLPs)is developed to solve the fractional order HIV/AIDS epidemic model with a treatment compartment.The HIV/AIDS mathematical model with a treatment compartment is divided into five classes,namely,susceptible patients(S),HIV-positive individuals(I),individuals with full-blown AIDS but not receiving ARV treatment(A),individuals being treated(T),and individuals who have changed their sexual habits sufficiently(R).According to the method,by utilizing the PLPs and the collocation points,we convert the fractional order HIV/AIDS epidemic model with a treatment compartment into a nonlinear system of the algebraic equations.Also,the error analysis is presented for the Pell-Lucas approximation method.The aim of this study is to observe the behavior of five populations after 200 days when drug treatment is applied to HIV-infectious and full-blown AIDS people.To demonstrate the usefulness of this method,the applications are made on the numerical example with the help of MATLAB.In addition,four cases of the fractional order derivative(p=1,p=0.95,p=0.9,p=0.85)are examined in the range[0,200].Owing to applications,we figured out that the outcomes have quite decent errors.Also,we understand that the errors decrease when the value of N increases.The figures in this study are created in MATLAB.The outcomes indicate that the presented method is reasonably sufficient and correct.展开更多
In this paper,we firstly recall some basic results on pseudo S-asymptotically(ω,c)-periodic functions and Sobolev type fractional differential equation.We secondly investigate some existence of pseudo S-asymptotical...In this paper,we firstly recall some basic results on pseudo S-asymptotically(ω,c)-periodic functions and Sobolev type fractional differential equation.We secondly investigate some existence of pseudo S-asymptotically(ω,c)-periodic solutions for a semilinear fractional differential equations of Sobolev type.We finally present a simple example.展开更多
Angle domain common imaging gathers(ADCIGs)serve as not only an ideal approach for tomographic velocity modeling but also as a crucial means of mitigating low-frequency noise.Thus,they play a significant role in seism...Angle domain common imaging gathers(ADCIGs)serve as not only an ideal approach for tomographic velocity modeling but also as a crucial means of mitigating low-frequency noise.Thus,they play a significant role in seismic data processing.Recently,the Poynting vector method,due to its lower computational requirements and higher resolution,has become a commonly used approach for obtaining ADCIGs.However,due to the viscoelastic properties of underground media,attenuation effects(phase dispersion and amplitude attenuation)have become a factor,which is important in seismic data processing.However,the primary applications of ADCIGs are currently confined to acoustic and elastic media.To assess the influence of attenuation and elastic effects on ADCIGs,we introduce an extraction method for ADCIGs based on fractional viscoelastic equations.This method enhances ADCIGs accuracy by simultaneously considering both the attenuation and elastic properties of underground media.Meanwhile,the S-wave quasi tensor is used to reduce the impact of P-wave energy on S-wave stress,thus further increasing the accuracy of PS-ADCIGs.In conclusion,our analysis examines the impact of the quality factor Q on ADCIGs and offers theoretical guidance for parameter inversion.展开更多
The numerical approach for finding the solution of fractional order systems of boundary value problems (BPVs) is derived in this paper. The implementation of the weighted residuals such as Galerkin, Least Square, and ...The numerical approach for finding the solution of fractional order systems of boundary value problems (BPVs) is derived in this paper. The implementation of the weighted residuals such as Galerkin, Least Square, and Collocation methods are included for solving fractional order differential equations, which is broadened to acquire the approximate solutions of fractional order systems with differentiable polynomials, namely Legendre polynomials, as basis functions. The algorithm of the residual formulations of matrix form can be coded efficiently. The interpretation of Caputo fractional derivatives is employed here. We have demonstrated these methods numerically through a few examples of linear and nonlinear BVPs. The results in absolute errors show that the present method efficiently finds the numerical solutions of fractional order systems of differential equations.展开更多
The higher-order numerical scheme of nonlinear advection-diffusion equations is studied in this article, where the space fractional derivatives are evaluated by using weighted and shifted Grünwald difference oper...The higher-order numerical scheme of nonlinear advection-diffusion equations is studied in this article, where the space fractional derivatives are evaluated by using weighted and shifted Grünwald difference operators and combining the compact technique, in the time direction is discretized by the Crank-Nicolson method. Through the energy method, the stability and convergence of the numerical scheme in the sense of L<sub>2</sub>-norm are proved, and the convergence order is . Some examples are given to show that our numerical scheme is effective.展开更多
In this paper, we focus on studying the fractional variational principle and the differential equations of motion for a fractional mechanical system. A combined Riemann-Liouville fractional derivative operator is defi...In this paper, we focus on studying the fractional variational principle and the differential equations of motion for a fractional mechanical system. A combined Riemann-Liouville fractional derivative operator is defined, and a fractional Hamilton principle under this definition is established. The fractional Lagrange equations and the fractional Hamilton canonical equations are derived from the fractional Hamilton principle. A number of special cases are given, showing the universality of our conclusions. At the end of the paper, an example is given to illustrate the application of the results.展开更多
In this paper,we study a Dirichlet-type boundary value problem(BVP) of nonlinear fractional differential equation with an order α∈(3,4],where the fractional derivative D~α_(o^+)is the standard Riemann-Liouville fra...In this paper,we study a Dirichlet-type boundary value problem(BVP) of nonlinear fractional differential equation with an order α∈(3,4],where the fractional derivative D~α_(o^+)is the standard Riemann-Liouville fractional derivative.By constructing the Green function and investigating its properties,we obtain some criteria for the existence of one positive solution and two positive solutions for the above BVP.The Krasnosel'skii fixedpoint theorem in cones is used here.We also give an example to illustrate the applicability of our results.展开更多
This paper is concerned with the boundary value problem of a nonlinear fractional differential equation. By means of Schauder fixed-point theorem, an existence result of solution is obtained.
The goal of this paper is to consider the global well-posedness to n-dimensional （n 〉 3） Boussinesq equations with fractional dissipation. More precisely, it is proved that there exists a unique global regular solu...The goal of this paper is to consider the global well-posedness to n-dimensional （n 〉 3） Boussinesq equations with fractional dissipation. More precisely, it is proved that there exists a unique global regular solution to the Boussinesq equations provided the real parameter α satisfies α≥1/2 ＋n/4.展开更多
This paper obtains Lagrange equations of nonholonomic systems with fractional derivatives. First, the exchanging relationships between the isochronous variation and the fractional derivatives are derived. Secondly, ba...This paper obtains Lagrange equations of nonholonomic systems with fractional derivatives. First, the exchanging relationships between the isochronous variation and the fractional derivatives are derived. Secondly, based on these exchanging relationships, the Hamilton＇s principle is presented for non-conservative systems with fractional derivatives. Thirdly, Lagrange equations of the systems are obtained. Furthermore, the d＇Alembert-Lagrange principle with fractional derivatives is presented, and the Lagrange equations of nonholonomic systems with fractional derivatives are studied. An example is designed to illustrate these results.展开更多
文摘Fractional-order time-delay differential equations can describe many complex physical phenomena with memory or delay effects, which are widely used in the fields of cell biology, control systems, signal processing, etc. Therefore, it is of great significance to study fractional-order time-delay differential equations. In this paper, we discuss a finite volume element method for a class of fractional-order neutral time-delay differential equations. By introducing an intermediate variable, the fourth-order problem is transformed into a system of equations consisting of two second-order partial differential equations. The L1 formula is used to approximate the time fractional order derivative terms, and the finite volume element method is used in space. A fully discrete format of the equations is established, and we prove the existence, uniqueness, convergence and stability of the solution. Finally, the validity of the format is verified by numerical examples.
文摘In this article, a finite volume element algorithm is presented and discussed for the numerical solutions of a time-fractional nonlinear fourth-order diffusion equation with time delay. By choosing the second-order spatial derivative of the original unknown as an additional variable, the fourth-order problem is transformed into a second-order system. Then the fully discrete finite volume element scheme is formulated by using L1approximation for temporal Caputo derivative and finite volume element method in spatial direction. The unique solvability and stable result of the proposed scheme are proved. A priori estimate of L2-norm with optimal order of convergence O(h2+τ2−α)where τand hare time step length and space mesh parameter, respectively, is obtained. The efficiency of the scheme is supported by some numerical experiments.
基金by the Subproject of Major Science and Technology Program of China(No.2017ZX07101001-01)the Fundamental Research Funds for the Central Universities(Nos.2018MS168 and 2020MS043).
文摘Anomalous diffusion is a widespread physical phenomenon,and numerical methods of fractional diffusion models are of important scientific significance and engineering application value.For time fractional diffusion-wave equation with damping,a difference(ASC-N,alternating segment Crank-Nicolson)scheme with intrinsic parallelism is proposed.Based on alternating technology,the ASC-N scheme is constructed with four kinds of Saul’yev asymmetric schemes and Crank-Nicolson(C-N)scheme.The unconditional stability and convergence are rigorously analyzed.The theoretical analysis and numerical experiments show that the ASC-N scheme is effective for solving time fractional diffusion-wave equation.
基金Supported by the National Natural Science Foundation of China(Nos.11771438,11471296)the Key Scientific Research Projects in Universities of Henan Province(No.19B110013)the Program for Scientific and Technological Innovation Talents in Universities of Henan Province(No.19HASTIT025)
文摘In this paper, high-order numerical analysis of finite element method（FEM） is presented for twodimensional multi-term time-fractional diffusion-wave equation（TFDWE）. First of all, a fully-discrete approximate scheme for multi-term TFDWE is established, which is based on bilinear FEM in spatial direction and Crank-Nicolson approximation in temporal direction, respectively. Then the proposed scheme is proved to be unconditionally stable and convergent. And then, rigorous proofs are given here for superclose properties in H-1-norm and temporal convergence in L-2-norm with order O（h-2＋ τ-（3-α））, where h and τ are the spatial size and time step, respectively. At the same time, theoretical analysis of global superconvergence in H-1-norm is derived by interpolation postprocessing technique. At last, numerical example is provided to demonstrate the theoretical analysis.
基金NSF of China[grant number:11371157]Natural Science Foundation of Anhui Higher Education Institutions of China[grant number:KJ2016A492]Natural Science Foundation of Bozhou College[grant number:BSKY201426,BSKY201535].
文摘In this paper,we study a new numerical technique for a class of 2D nonlinear fractional diffusion-wave equations with the Caputo-type temporal derivative and Riesz-type spatial derivative.Galerkin finite element scheme is used for the discretization in the spatial direction,and the temporal component is discretized by a new alternating direction implicit(ADI)method.Next,we strictly prove that the numerical method is stable and convergent.Finally,to confirm our theoretical analysis,some numerical examples in 2D space are presented.
文摘This article describes the solution procedure of the fractional Pade-Ⅱ equation and generalized Zakharov equation(GSEs)using the sine-cosine method.Pade-Ⅱ is an important nonlinear wave equation modeling unidirectional propagation of long-wave in dispersive media and GSEs are used to model the interaction between one-dimensional high,and low-frequency waves.Classes of trigonometric and hyperbolic function solutions in fractional calculus are discussed.Graphical simulations of the numerical solutions are flaunted by MATLAB.
基金Project supported by the National Natural Science Foun dation of China(Grant No.11274398).
文摘A significant obstacle impeding the advancement of the time fractional Schrodinger equation lies in the challenge of determining its precise mathematical formulation.In order to address this,we undertake an exploration of the time fractional Schrodinger equation within the context of a non-Markovian environment.By leveraging a two-level atom as an illustrative case,we find that the choice to raise i to the order of the time derivative is inappropriate.In contrast to the conventional approach used to depict the dynamic evolution of quantum states in a non-Markovian environment,the time fractional Schrodinger equation,when devoid of fractional-order operations on the imaginary unit i,emerges as a more intuitively comprehensible framework in physics and offers greater simplicity in computational aspects.Meanwhile,we also prove that it is meaningless to study the memory of time fractional Schrodinger equation with time derivative 1<α≤2.It should be noted that we have not yet constructed an open system that can be fully described by the time fractional Schrodinger equation.This will be the focus of future research.Our study might provide a new perspective on the role of time fractional Schrodinger equation.
基金supported by National Natural Science Foundation of China(12271277)the Open Research Fund of Key Laboratory of Nonlinear Analysis&Applications(Central China Normal University),Ministry of Education,China.
文摘In this article,we consider the diffusion equation with multi-term time-fractional derivatives.We first derive,by a subordination principle for the solution,that the solution is positive when the initial value is non-negative.As an application,we prove the uniqueness of solution to an inverse problem of determination of the temporally varying source term by integral type information in a subdomain.Finally,several numerical experiments are presented to show the accuracy and efficiency of the algorithm.
基金the National Key Research and Development Program of China(No.2021YFB2600704)the National Natural Science Foundation of China(No.52171272)the Significant Science and Technology Project of the Ministry of Water Resources of China(No.SKS-2022112).
文摘Anovel accuratemethod is proposed to solve a broad variety of linear and nonlinear(1+1)-dimensional and(2+1)-dimensional multi-term time-fractional partial differential equations with spatial operators of anisotropic diffusivity.For(1+1)-dimensional problems,analytical solutions that satisfy the boundary requirements are derived.Such solutions are numerically calculated using the trigonometric basis approximation for(2+1)-dimensional problems.With the aid of these analytical or numerical approximations,the original problems can be converted into the fractional ordinary differential equations,and solutions to the fractional ordinary differential equations are approximated by modified radial basis functions with time-dependent coefficients.An efficient backward substitution strategy that was previously provided for a single fractional ordinary differential equation is then used to solve the corresponding systems.The straightforward quasilinearization technique is applied to handle nonlinear issues.Numerical experiments demonstrate the suggested algorithm’s superior accuracy and efficiency.
基金supported by the Science and Technology Development Fund of Macao SAR(FDCT0128/2022/A,0020/2023/RIB1,0111/2023/AFJ,005/2022/ALC)the Shandong Natural Science Foundation of China(ZR2020MA004)+2 种基金the National Natural Science Foundation of China(12071272)the MYRG 2018-00168-FSTZhejiang Provincial Natural Science Foundation of China(LQ23A010014).
文摘This study introduces a pre-orthogonal adaptive Fourier decomposition(POAFD)to obtain approximations and numerical solutions to the fractional Laplacian initial value problem and the extension problem of Caffarelli and Silvestre(generalized Poisson equation).As a first step,the method expands the initial data function into a sparse series of the fundamental solutions with fast convergence,and,as a second step,makes use of the semigroup or the reproducing kernel property of each of the expanding entries.Experiments show the effectiveness and efficiency of the proposed series solutions.
文摘In this study,a numerical method based on the Pell-Lucas polynomials(PLPs)is developed to solve the fractional order HIV/AIDS epidemic model with a treatment compartment.The HIV/AIDS mathematical model with a treatment compartment is divided into five classes,namely,susceptible patients(S),HIV-positive individuals(I),individuals with full-blown AIDS but not receiving ARV treatment(A),individuals being treated(T),and individuals who have changed their sexual habits sufficiently(R).According to the method,by utilizing the PLPs and the collocation points,we convert the fractional order HIV/AIDS epidemic model with a treatment compartment into a nonlinear system of the algebraic equations.Also,the error analysis is presented for the Pell-Lucas approximation method.The aim of this study is to observe the behavior of five populations after 200 days when drug treatment is applied to HIV-infectious and full-blown AIDS people.To demonstrate the usefulness of this method,the applications are made on the numerical example with the help of MATLAB.In addition,four cases of the fractional order derivative(p=1,p=0.95,p=0.9,p=0.85)are examined in the range[0,200].Owing to applications,we figured out that the outcomes have quite decent errors.Also,we understand that the errors decrease when the value of N increases.The figures in this study are created in MATLAB.The outcomes indicate that the presented method is reasonably sufficient and correct.
基金supported by NSF of Shaanxi Province(Grant No.2023-JC-YB-011).
文摘In this paper,we firstly recall some basic results on pseudo S-asymptotically(ω,c)-periodic functions and Sobolev type fractional differential equation.We secondly investigate some existence of pseudo S-asymptotically(ω,c)-periodic solutions for a semilinear fractional differential equations of Sobolev type.We finally present a simple example.
基金supported by the National Natural Science Foundation of China(NSFC)under contract number 42274147 and 41874144。
文摘Angle domain common imaging gathers(ADCIGs)serve as not only an ideal approach for tomographic velocity modeling but also as a crucial means of mitigating low-frequency noise.Thus,they play a significant role in seismic data processing.Recently,the Poynting vector method,due to its lower computational requirements and higher resolution,has become a commonly used approach for obtaining ADCIGs.However,due to the viscoelastic properties of underground media,attenuation effects(phase dispersion and amplitude attenuation)have become a factor,which is important in seismic data processing.However,the primary applications of ADCIGs are currently confined to acoustic and elastic media.To assess the influence of attenuation and elastic effects on ADCIGs,we introduce an extraction method for ADCIGs based on fractional viscoelastic equations.This method enhances ADCIGs accuracy by simultaneously considering both the attenuation and elastic properties of underground media.Meanwhile,the S-wave quasi tensor is used to reduce the impact of P-wave energy on S-wave stress,thus further increasing the accuracy of PS-ADCIGs.In conclusion,our analysis examines the impact of the quality factor Q on ADCIGs and offers theoretical guidance for parameter inversion.
文摘The numerical approach for finding the solution of fractional order systems of boundary value problems (BPVs) is derived in this paper. The implementation of the weighted residuals such as Galerkin, Least Square, and Collocation methods are included for solving fractional order differential equations, which is broadened to acquire the approximate solutions of fractional order systems with differentiable polynomials, namely Legendre polynomials, as basis functions. The algorithm of the residual formulations of matrix form can be coded efficiently. The interpretation of Caputo fractional derivatives is employed here. We have demonstrated these methods numerically through a few examples of linear and nonlinear BVPs. The results in absolute errors show that the present method efficiently finds the numerical solutions of fractional order systems of differential equations.
文摘The higher-order numerical scheme of nonlinear advection-diffusion equations is studied in this article, where the space fractional derivatives are evaluated by using weighted and shifted Grünwald difference operators and combining the compact technique, in the time direction is discretized by the Crank-Nicolson method. Through the energy method, the stability and convergence of the numerical scheme in the sense of L<sub>2</sub>-norm are proved, and the convergence order is . Some examples are given to show that our numerical scheme is effective.
基金Project supported by the National Natural Science Foundation of China (Grant No. 10972151)
文摘In this paper, we focus on studying the fractional variational principle and the differential equations of motion for a fractional mechanical system. A combined Riemann-Liouville fractional derivative operator is defined, and a fractional Hamilton principle under this definition is established. The fractional Lagrange equations and the fractional Hamilton canonical equations are derived from the fractional Hamilton principle. A number of special cases are given, showing the universality of our conclusions. At the end of the paper, an example is given to illustrate the application of the results.
基金Supported by the Research Fund for the Doctoral Program of High Education of China(20094407110001)Supported by the NSF of Guangdong Province(10151063101000003)
文摘In this paper,we study a Dirichlet-type boundary value problem(BVP) of nonlinear fractional differential equation with an order α∈(3,4],where the fractional derivative D~α_(o^+)is the standard Riemann-Liouville fractional derivative.By constructing the Green function and investigating its properties,we obtain some criteria for the existence of one positive solution and two positive solutions for the above BVP.The Krasnosel'skii fixedpoint theorem in cones is used here.We also give an example to illustrate the applicability of our results.
文摘This paper is concerned with the boundary value problem of a nonlinear fractional differential equation. By means of Schauder fixed-point theorem, an existence result of solution is obtained.
基金partially supported by NSFC(1117102611371059)+1 种基金BNSF(2112023)the Fundamental Research Funds for the Central Universities of China
文摘The goal of this paper is to consider the global well-posedness to n-dimensional （n 〉 3） Boussinesq equations with fractional dissipation. More precisely, it is proved that there exists a unique global regular solution to the Boussinesq equations provided the real parameter α satisfies α≥1/2 ＋n/4.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 11072218 and 10672143)
文摘This paper obtains Lagrange equations of nonholonomic systems with fractional derivatives. First, the exchanging relationships between the isochronous variation and the fractional derivatives are derived. Secondly, based on these exchanging relationships, the Hamilton＇s principle is presented for non-conservative systems with fractional derivatives. Thirdly, Lagrange equations of the systems are obtained. Furthermore, the d＇Alembert-Lagrange principle with fractional derivatives is presented, and the Lagrange equations of nonholonomic systems with fractional derivatives are studied. An example is designed to illustrate these results.
