In this paper,a local discontinuous Galerkin(LDG)scheme for the time-fractional diffusion equation is proposed and analyzed.The Caputo time-fractional derivative(of orderα,with 0<α<1)is approximated by a finit...In this paper,a local discontinuous Galerkin(LDG)scheme for the time-fractional diffusion equation is proposed and analyzed.The Caputo time-fractional derivative(of orderα,with 0<α<1)is approximated by a finite difference method with an accuracy of order3-α,and the space discretization is based on the LDG method.For the finite difference method,we summarize and supplement some previous work by others,and apply it to the analysis of the convergence and stability of the proposed scheme.The optimal error estimate is obtained in the L2norm,indicating that the scheme has temporal(3-α)th-order accuracy and spatial(k+1)th-order accuracy,where k denotes the highest degree of a piecewise polynomial in discontinuous finite element space.The numerical results are also provided to verify the accuracy and efficiency of the considered scheme.展开更多
Fractional diffusion equations have been the focus of modeling problems in hydrology, biology, viscoelasticity, physics, engineering, and other areas of applications. In this paper, a meshfree method based on the movi...Fractional diffusion equations have been the focus of modeling problems in hydrology, biology, viscoelasticity, physics, engineering, and other areas of applications. In this paper, a meshfree method based on the moving Kriging inter- polation is developed for a two-dimensional time-fractional diffusion equation. The shape function and its derivatives are obtained by the moving Kriging interpolation technique. For possessing the Kronecker delta property, this technique is very efficient in imposing the essential boundary conditions. The governing time-fractional diffusion equations are transformed into a standard weak formulation by the Galerkin method. It is then discretized into a meshfree system of time-dependent equations, which are solved by the standard central difference method. Numerical examples illustrating the applicability and effectiveness of the proposed method are presented and discussed in detail.展开更多
This article proposes a high-order numerical method for a space distributed-order time-fractional diffusion equation.First,we use the mid-point quadrature rule to transform the space distributed-order term into multi-...This article proposes a high-order numerical method for a space distributed-order time-fractional diffusion equation.First,we use the mid-point quadrature rule to transform the space distributed-order term into multi-term fractional derivatives.Second,based on the piecewise-quadratic polynomials,we construct the nodal basis functions,and then discretize the multi-term fractional equation by the finite volume method.For the time-fractional derivative,the finite difference method is used.Finally,the iterative scheme is proved to be unconditionally stable and convergent with the accuracy O(σ^(2)+τ^(2-β)+h^(3)),whereτand h are the time step size and the space step size,respectively.A numerical example is presented to verify the effectiveness of the proposed method.展开更多
In this paper,we consider the inverse problem for identifying the source term of the time-fractional equation with a hyper-Bessel operator.First,we prove that this inverse problem is ill-posed,and give the conditional...In this paper,we consider the inverse problem for identifying the source term of the time-fractional equation with a hyper-Bessel operator.First,we prove that this inverse problem is ill-posed,and give the conditional stability.Then,we give the optimal error bound for this inverse problem.Next,we use the fractional Tikhonov regularization method and the fractional Landweber iterative regularization method to restore the stability of the ill-posed problem,and give corresponding error estimates under different regularization parameter selection rules.Finally,we verify the effectiveness of the method through numerical examples.展开更多
The variational iteration method is successfully extended to the case of solving fractional differential equations, and the Lagrange multiplier of the method is identified in a more accurate way. Some diffusion models...The variational iteration method is successfully extended to the case of solving fractional differential equations, and the Lagrange multiplier of the method is identified in a more accurate way. Some diffusion models with fractional derivatives are investigated analytically, and the results show the efficiency of the new Lagrange multiplier for fractional differential equations of arbitrary order.展开更多
Innovative definitions of the electric and magnetic diffusivities through conducting mediums and innovative diffusion equations of the electric charges and magnetic flux are verified in this article. Such innovations ...Innovative definitions of the electric and magnetic diffusivities through conducting mediums and innovative diffusion equations of the electric charges and magnetic flux are verified in this article. Such innovations depend on the analogy of the governing laws of diffusion of the thermal, electrical, and magnetic energies and newly defined natures of the electric charges and magnetic flux as energy, or as electromagnetic waves, that have electric and magnetic potentials. The introduced diffusion equations of the electric charges and magnetic flux involve Laplacian operator and the introduced diffusivities. Both equations are applied to determine the electric and magnetic fields in conductors as the heat diffusion equation which is applied to determine the thermal field in steady and unsteady heat diffusion conditions. The use of electric networks for experimental modeling of thermal networks represents sufficient proof of similarity of the diffusion equations of both fields. By analysis of the diffusion phenomena of the three considered modes of energy transfer;the rates of flow of these energies are found to be directly proportional to the gradient of their volumetric concentration, or density, and the proportionality constants in such relations are the diffusivity of each energy. Such analysis leads also to find proportionality relations between the potentials of such energies and their volumetric concentrations. Validity of the introduced diffusion equations is verified by correspondence their solutions to the measurement results of the electric and magnetic fields in microwave ovens.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,an inverse source problem for the time-fractional diffusion equation is investigated.The observational data is on the final time and the source term is assumed to be temporally independent and with a spa...In this paper,an inverse source problem for the time-fractional diffusion equation is investigated.The observational data is on the final time and the source term is assumed to be temporally independent and with a sparse structure.Here the sparsity is understood with respect to the pixel basis,i.e.,the source has a small support.By an elastic-net regularization method,this inverse source problem is formulated into an optimization problem and a semismooth Newton(SSN)algorithm is developed to solve it.A discretization strategy is applied in the numerical realization.Several one and two dimensional numerical examples illustrate the efficiency of the proposed method.展开更多
In this paper,we study the well-posedness and solution regularity of a multi-term variable-order time-fractional diffusion equation,and then develop an optimal Galerkin finite element scheme without any regularity ass...In this paper,we study the well-posedness and solution regularity of a multi-term variable-order time-fractional diffusion equation,and then develop an optimal Galerkin finite element scheme without any regularity assumption on its true solution.We show that the solution regularity of the considered problem can be affected by the maximum value of variable-order at initial time t=0.More precisely,we prove that the solution to the multi-term variable-order time-fractional diffusion equation belongs to C 2([0,T])in time provided that the maximum value has an integer limit near the initial time and the data has sufficient smoothness,otherwise the solution exhibits the same singular behavior like its constant-order counterpart.Based on these regularity results,we prove optimalorder convergence rate of the Galerkin finite element scheme.Furthermore,we develop an efficient parallel-in-time algorithm to reduce the computational costs of the evaluation of multi-term variable-order fractional derivatives.Numerical experiments are put forward to verify the theoretical findings and to demonstrate the efficiency of the proposed scheme.展开更多
In this paper, a fully discrete scheme based on the LI approximation in temporal direction for the fractional derivative of order in (0,1) and nonconforming mixed finite element method (MFEM) in spatial direction is e...In this paper, a fully discrete scheme based on the LI approximation in temporal direction for the fractional derivative of order in (0,1) and nonconforming mixed finite element method (MFEM) in spatial direction is established. First, we prove a novel result of the consistency error estimate with order O(h^2)of EQ1^rot element (see Lemma 2.3). Then, by using the proved character of EQ1^rot element, we present the superconvergent estimates for the original variable u in the broken H^1-norm and the flux →p =△u in the (L^2)^2-norm under a weaker regularity of the exact solution. Finally, numerical results are provided to confirm the theoretical analysis.展开更多
In this paper,a new type of the discrete fractional Gronwall inequality is developed,which is applied to analyze the stability and convergence of a Galerkin spectral method for a linear time-fractional subdifiFusion e...In this paper,a new type of the discrete fractional Gronwall inequality is developed,which is applied to analyze the stability and convergence of a Galerkin spectral method for a linear time-fractional subdifiFusion equation.Based on the temporal-spatial error splitting argument technique,the discrete fractional Gronwall inequality is also applied to prove the unconditional convergence of a semi-implicit Galerkin spectral method for a nonlinear time-fractional subdififusion equation.展开更多
This paper is aimed at solving the nonlinear time-fractional partial differential equation with two small parameters arising from option pricing model in financial economics.The traditional reproducing kernel(RK)metho...This paper is aimed at solving the nonlinear time-fractional partial differential equation with two small parameters arising from option pricing model in financial economics.The traditional reproducing kernel(RK)method which deals with this problem is very troublesome.This paper proposes a new method by adaptive multi-step piecewise interpolation reproducing kernel(AMPIRK)method for the first time.This method has three obvious advantages which are as follows.Firstly,the piecewise number is reduced.Secondly,the calculation accuracy is improved.Finally,the waste time caused by too many fragments is avoided.Then four numerical examples show that this new method has a higher precision and it is a more timesaving numerical method than the others.The research in this paper provides a powerful mathematical tool for solving time-fractional option pricing model which will play an important role in financial economics.展开更多
The analytical solution of the multi-dimensional,time-fractional model of Navier-Stokes equation using the triple and quadruple Elzaki transformdecompositionmethod is presented in this article.The aforesaidmodel is an...The analytical solution of the multi-dimensional,time-fractional model of Navier-Stokes equation using the triple and quadruple Elzaki transformdecompositionmethod is presented in this article.The aforesaidmodel is analyzed by employing Caputo fractional derivative.We deliberated three stimulating examples that correspond to the triple and quadruple Elzaki transform decomposition methods,respectively.The findings illustrate that the established approaches are extremely helpful in obtaining exact and approximate solutions to the problems.The exact and estimated solutions are delineated via numerical simulation.The proposed analysis indicates that the projected configuration is extremely meticulous,highly efficient,and precise in understanding the behavior of complex evolutionary problems of both fractional and integer order that classify affiliated scientific fields and technology.展开更多
The time-fractional modified Korteweg-de Vries(KdV)equation is committed to establish exact solutions by employing the bifurcation method.Firstly,the phase portraits and related qualitative analysis are comprehensivel...The time-fractional modified Korteweg-de Vries(KdV)equation is committed to establish exact solutions by employing the bifurcation method.Firstly,the phase portraits and related qualitative analysis are comprehensively provided.Then,we give parametric expressions of different types of solutions matching with the corresponding orbits.Finally,solution profiles,3D and density plots of some solutions are presented with proper parametric choices.展开更多
We present a mathematical and numerical study for a pointwise optimal control problem governed by a variable-coefficient Riesz-fractional diffusion equation.Due to the impact of the variable diffusivity coefficient,ex...We present a mathematical and numerical study for a pointwise optimal control problem governed by a variable-coefficient Riesz-fractional diffusion equation.Due to the impact of the variable diffusivity coefficient,existing regularity results for their constantcoefficient counterparts do not apply,while the bilinear forms of the state(adjoint)equation may lose the coercivity that is critical in error estimates of the finite element method.We reformulate the state equation as an equivalent constant-coefficient fractional diffusion equation with the addition of a variable-coefficient low-order fractional advection term.First order optimality conditions are accordingly derived and the smoothing properties of the solutions are analyzed by,e.g.,interpolation estimates.The weak coercivity of the resulting bilinear forms are proven via the Garding inequality,based on which we prove the optimal-order convergence estimates of the finite element method for the(adjoint)state variable and the control variable.Numerical experiments substantiate the theoretical predictions.展开更多
In this paper,finite difference schemes for solving time-space fractional diffusion equations in one dimension and two dimensions are proposed.The temporal derivative is in the Caputo-Hadamard sense for both cases.The...In this paper,finite difference schemes for solving time-space fractional diffusion equations in one dimension and two dimensions are proposed.The temporal derivative is in the Caputo-Hadamard sense for both cases.The spatial derivative for the one-dimensional equation is of Riesz definition and the two-dimensional spatial derivative is given by the fractional Laplacian.The schemes are proved to be unconditionally stable and convergent.The numerical results are in line with the theoretical analysis.展开更多
For reaction-diffusion equations in irregular domains with moving boundaries,the numerical stability constraints from the reaction and diffusion terms often require very restricted time step sizes,while complex geomet...For reaction-diffusion equations in irregular domains with moving boundaries,the numerical stability constraints from the reaction and diffusion terms often require very restricted time step sizes,while complex geometries may lead to difficulties in the accuracy when discretizing the high-order derivatives on grid points near the boundary.It is very challenging to design numerical methods that can efficiently and accurately handle both difficulties.Applying an implicit scheme may be able to remove the stability constraints on the time step,however,it usually requires solving a large global system of nonlinear equations for each time step,and the computational cost could be significant.Integration factor(IF)or exponential time differencing(ETD)methods are one of the popular methods for temporal partial differential equations(PDEs)among many other methods.In our paper,we couple ETD methods with an embedded boundary method to solve a system of reaction-diffusion equations with complex geometries.In particular,we rewrite all ETD schemes into a linear combination of specificФ-functions and apply one state-of-the-art algorithm to compute the matrix-vector multiplications,which offers significant computational advantages with adaptive Krylov subspaces.In addition,we extend this method by incorporating the level set method to solve the free boundary problem.The accuracy,stability,and efficiency of the developed method are demonstrated by numerical examples.展开更多
A class of singularly perturbed initial boundary value problems for the reaction diffusion equations in a part of domain are considered. Using the operator theory the asymptotic behavior of solution for the problems i...A class of singularly perturbed initial boundary value problems for the reaction diffusion equations in a part of domain are considered. Using the operator theory the asymptotic behavior of solution for the problems is studied.展开更多
This paper is concerned with the asymptotic behavior of solutions to the initial boundary problem of the two-dimensional density-dependent Boussinesq equations.It is shown that the solutions of the Boussinesq equation...This paper is concerned with the asymptotic behavior of solutions to the initial boundary problem of the two-dimensional density-dependent Boussinesq equations.It is shown that the solutions of the Boussinesq equations converge to those of zero thermal diffusivity Boussinesq equations as the thermal diffusivity tends to zero,and the convergence rate is established.In addition,we prove that the boundary-layer thickness is of the valueδ(k)=k^(α)with anyα∈(0,1/4)for a small diffusivity coefficient k>0,and we also find a function to describe the properties of the boundary layer.展开更多
A class of nonlinear singularly perturbed problems for reaction diffusion equations are considered. Under suitable conditions, by using the theory of differential inequalities, the asymptotic behavior of solutions for...A class of nonlinear singularly perturbed problems for reaction diffusion equations are considered. Under suitable conditions, by using the theory of differential inequalities, the asymptotic behavior of solutions for the initial boundary value problems are studied, reduced problems of which possess two intersecting solutions.展开更多
基金supported by the State Key Program of National Natural Science Foundation of China(11931003)the National Natural Science Foundation of China(41974133)。
文摘In this paper,a local discontinuous Galerkin(LDG)scheme for the time-fractional diffusion equation is proposed and analyzed.The Caputo time-fractional derivative(of orderα,with 0<α<1)is approximated by a finite difference method with an accuracy of order3-α,and the space discretization is based on the LDG method.For the finite difference method,we summarize and supplement some previous work by others,and apply it to the analysis of the convergence and stability of the proposed scheme.The optimal error estimate is obtained in the L2norm,indicating that the scheme has temporal(3-α)th-order accuracy and spatial(k+1)th-order accuracy,where k denotes the highest degree of a piecewise polynomial in discontinuous finite element space.The numerical results are also provided to verify the accuracy and efficiency of the considered scheme.
基金Project supported by the National Natural Science Foundation of China(Grant No.11072117)the Natural Science Foundation of Ningbo City,China(GrantNo.2013A610103)+2 种基金the Natural Science Foundation of Zhejiang Province,China(Grant No.Y6090131)the Disciplinary Project of Ningbo City,China(GrantNo.SZXL1067)the K.C.Wong Magna Fund in Ningbo University,China
文摘Fractional diffusion equations have been the focus of modeling problems in hydrology, biology, viscoelasticity, physics, engineering, and other areas of applications. In this paper, a meshfree method based on the moving Kriging inter- polation is developed for a two-dimensional time-fractional diffusion equation. The shape function and its derivatives are obtained by the moving Kriging interpolation technique. For possessing the Kronecker delta property, this technique is very efficient in imposing the essential boundary conditions. The governing time-fractional diffusion equations are transformed into a standard weak formulation by the Galerkin method. It is then discretized into a meshfree system of time-dependent equations, which are solved by the standard central difference method. Numerical examples illustrating the applicability and effectiveness of the proposed method are presented and discussed in detail.
基金supported by the Natural and Science Foundation Council of China(11771059)Hunan Provincial Natural Science Foundation of China(2018JJ3519)Scientific Research Project of Hunan Provincial office of Education(20A022)。
文摘This article proposes a high-order numerical method for a space distributed-order time-fractional diffusion equation.First,we use the mid-point quadrature rule to transform the space distributed-order term into multi-term fractional derivatives.Second,based on the piecewise-quadratic polynomials,we construct the nodal basis functions,and then discretize the multi-term fractional equation by the finite volume method.For the time-fractional derivative,the finite difference method is used.Finally,the iterative scheme is proved to be unconditionally stable and convergent with the accuracy O(σ^(2)+τ^(2-β)+h^(3)),whereτand h are the time step size and the space step size,respectively.A numerical example is presented to verify the effectiveness of the proposed method.
基金supported by the National Natural Science Foundation of China(11961044)the Doctor Fund of Lan Zhou University of Technologythe Natural Science Foundation of Gansu Provice(21JR7RA214)。
文摘In this paper,we consider the inverse problem for identifying the source term of the time-fractional equation with a hyper-Bessel operator.First,we prove that this inverse problem is ill-posed,and give the conditional stability.Then,we give the optimal error bound for this inverse problem.Next,we use the fractional Tikhonov regularization method and the fractional Landweber iterative regularization method to restore the stability of the ill-posed problem,and give corresponding error estimates under different regularization parameter selection rules.Finally,we verify the effectiveness of the method through numerical examples.
基金Project supported by the Key Program of the National Natural Science Foundation of China (Grant No. 51134018).
文摘The variational iteration method is successfully extended to the case of solving fractional differential equations, and the Lagrange multiplier of the method is identified in a more accurate way. Some diffusion models with fractional derivatives are investigated analytically, and the results show the efficiency of the new Lagrange multiplier for fractional differential equations of arbitrary order.
文摘Innovative definitions of the electric and magnetic diffusivities through conducting mediums and innovative diffusion equations of the electric charges and magnetic flux are verified in this article. Such innovations depend on the analogy of the governing laws of diffusion of the thermal, electrical, and magnetic energies and newly defined natures of the electric charges and magnetic flux as energy, or as electromagnetic waves, that have electric and magnetic potentials. The introduced diffusion equations of the electric charges and magnetic flux involve Laplacian operator and the introduced diffusivities. Both equations are applied to determine the electric and magnetic fields in conductors as the heat diffusion equation which is applied to determine the thermal field in steady and unsteady heat diffusion conditions. The use of electric networks for experimental modeling of thermal networks represents sufficient proof of similarity of the diffusion equations of both fields. By analysis of the diffusion phenomena of the three considered modes of energy transfer;the rates of flow of these energies are found to be directly proportional to the gradient of their volumetric concentration, or density, and the proportionality constants in such relations are the diffusivity of each energy. Such analysis leads also to find proportionality relations between the potentials of such energies and their volumetric concentrations. Validity of the introduced diffusion equations is verified by correspondence their solutions to the measurement results of the electric and magnetic fields in microwave ovens.
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supported by National Science Foundation of China No.11171305 and No.91230203 and the work of X.Lu is partially supported by National Science Foundation of China No.11471253,the Fundamental Research Funds for the Central Universities(13lgzd07)and the PSTNS of Zhu Jiang in Guangzhou city(2011J2200099).
文摘In this paper,an inverse source problem for the time-fractional diffusion equation is investigated.The observational data is on the final time and the source term is assumed to be temporally independent and with a sparse structure.Here the sparsity is understood with respect to the pixel basis,i.e.,the source has a small support.By an elastic-net regularization method,this inverse source problem is formulated into an optimization problem and a semismooth Newton(SSN)algorithm is developed to solve it.A discretization strategy is applied in the numerical realization.Several one and two dimensional numerical examples illustrate the efficiency of the proposed method.
基金the National Natural Science Foundation of China(No.11971482)the Natural Science Foundation of Shandong Province(No.ZR2017MA006)+2 种基金the National Science Foundation(No.DMS-1620194)the China Postdoctoral Science Foundation(Nos.2020M681136,2021TQ0017,2021T140129)the International Postdoctoral Exchange Fellowship Program(Talent-Introduction Program)(No.YJ20210019).
文摘In this paper,we study the well-posedness and solution regularity of a multi-term variable-order time-fractional diffusion equation,and then develop an optimal Galerkin finite element scheme without any regularity assumption on its true solution.We show that the solution regularity of the considered problem can be affected by the maximum value of variable-order at initial time t=0.More precisely,we prove that the solution to the multi-term variable-order time-fractional diffusion equation belongs to C 2([0,T])in time provided that the maximum value has an integer limit near the initial time and the data has sufficient smoothness,otherwise the solution exhibits the same singular behavior like its constant-order counterpart.Based on these regularity results,we prove optimalorder convergence rate of the Galerkin finite element scheme.Furthermore,we develop an efficient parallel-in-time algorithm to reduce the computational costs of the evaluation of multi-term variable-order fractional derivatives.Numerical experiments are put forward to verify the theoretical findings and to demonstrate the efficiency of the proposed scheme.
基金the National Natural Science Foundation of China (No. 1167136911271340).
文摘In this paper, a fully discrete scheme based on the LI approximation in temporal direction for the fractional derivative of order in (0,1) and nonconforming mixed finite element method (MFEM) in spatial direction is established. First, we prove a novel result of the consistency error estimate with order O(h^2)of EQ1^rot element (see Lemma 2.3). Then, by using the proved character of EQ1^rot element, we present the superconvergent estimates for the original variable u in the broken H^1-norm and the flux →p =△u in the (L^2)^2-norm under a weaker regularity of the exact solution. Finally, numerical results are provided to confirm the theoretical analysis.
文摘In this paper,a new type of the discrete fractional Gronwall inequality is developed,which is applied to analyze the stability and convergence of a Galerkin spectral method for a linear time-fractional subdifiFusion equation.Based on the temporal-spatial error splitting argument technique,the discrete fractional Gronwall inequality is also applied to prove the unconditional convergence of a semi-implicit Galerkin spectral method for a nonlinear time-fractional subdififusion equation.
基金the National Natural Science Foundation of China(Grant Nos.71961022,11902163,12265020,and 12262024)the Natural Science Foundation of Inner Mongolia Autonomous Region of China(Grant Nos.2019BS01011 and 2022MS01003)+5 种基金2022 Inner Mongolia Autonomous Region Grassland Talents Project-Young Innovative and Entrepreneurial Talents(Mingjing Du)2022 Talent Development Foundation of Inner Mongolia Autonomous Region of China(Ming-Jing Du)the Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region Program(Grant No.NJYT-20-B18)the Key Project of High-quality Economic Development Research Base of Yellow River Basin in 2022(Grant No.21HZD03)2022 Inner Mongolia Autonomous Region International Science and Technology Cooperation High-end Foreign Experts Introduction Project(Ge Kai)MOE(Ministry of Education in China)Humanities and Social Sciences Foundation(Grants No.20YJC860005).
文摘This paper is aimed at solving the nonlinear time-fractional partial differential equation with two small parameters arising from option pricing model in financial economics.The traditional reproducing kernel(RK)method which deals with this problem is very troublesome.This paper proposes a new method by adaptive multi-step piecewise interpolation reproducing kernel(AMPIRK)method for the first time.This method has three obvious advantages which are as follows.Firstly,the piecewise number is reduced.Secondly,the calculation accuracy is improved.Finally,the waste time caused by too many fragments is avoided.Then four numerical examples show that this new method has a higher precision and it is a more timesaving numerical method than the others.The research in this paper provides a powerful mathematical tool for solving time-fractional option pricing model which will play an important role in financial economics.
基金supported by the Natural Science Foundation of China(GrantNos.61673169,11301127,11701176,11626101,11601485).
文摘The analytical solution of the multi-dimensional,time-fractional model of Navier-Stokes equation using the triple and quadruple Elzaki transformdecompositionmethod is presented in this article.The aforesaidmodel is analyzed by employing Caputo fractional derivative.We deliberated three stimulating examples that correspond to the triple and quadruple Elzaki transform decomposition methods,respectively.The findings illustrate that the established approaches are extremely helpful in obtaining exact and approximate solutions to the problems.The exact and estimated solutions are delineated via numerical simulation.The proposed analysis indicates that the projected configuration is extremely meticulous,highly efficient,and precise in understanding the behavior of complex evolutionary problems of both fractional and integer order that classify affiliated scientific fields and technology.
基金Project supported by the Natural Science Foundation of Shandong Province (Grant No.ZR2021MA084)the Natural Science Foundation of Liaocheng University (Grant No.318012025)Discipline with Strong Characteristics of Liaocheng University–Intelligent Science and Technology (Grant No.319462208)。
文摘The time-fractional modified Korteweg-de Vries(KdV)equation is committed to establish exact solutions by employing the bifurcation method.Firstly,the phase portraits and related qualitative analysis are comprehensively provided.Then,we give parametric expressions of different types of solutions matching with the corresponding orbits.Finally,solution profiles,3D and density plots of some solutions are presented with proper parametric choices.
基金supported by the National Natural Science Foundation of China(11971276,12171287)Natural Science Foundation of Shandong Province(ZR2016JL004)+1 种基金supported by the China Postdoctoral Science Foundation(2021TQ0017,2021M700244)International Postdoctoral Exchange Fellowship Program(Talent-Introduction Program)(YJ20210019)。
文摘We present a mathematical and numerical study for a pointwise optimal control problem governed by a variable-coefficient Riesz-fractional diffusion equation.Due to the impact of the variable diffusivity coefficient,existing regularity results for their constantcoefficient counterparts do not apply,while the bilinear forms of the state(adjoint)equation may lose the coercivity that is critical in error estimates of the finite element method.We reformulate the state equation as an equivalent constant-coefficient fractional diffusion equation with the addition of a variable-coefficient low-order fractional advection term.First order optimality conditions are accordingly derived and the smoothing properties of the solutions are analyzed by,e.g.,interpolation estimates.The weak coercivity of the resulting bilinear forms are proven via the Garding inequality,based on which we prove the optimal-order convergence estimates of the finite element method for the(adjoint)state variable and the control variable.Numerical experiments substantiate the theoretical predictions.
基金the National Natural Science Foundation of China under Grant Nos.12271339 and 12201391.
文摘In this paper,finite difference schemes for solving time-space fractional diffusion equations in one dimension and two dimensions are proposed.The temporal derivative is in the Caputo-Hadamard sense for both cases.The spatial derivative for the one-dimensional equation is of Riesz definition and the two-dimensional spatial derivative is given by the fractional Laplacian.The schemes are proved to be unconditionally stable and convergent.The numerical results are in line with the theoretical analysis.
文摘For reaction-diffusion equations in irregular domains with moving boundaries,the numerical stability constraints from the reaction and diffusion terms often require very restricted time step sizes,while complex geometries may lead to difficulties in the accuracy when discretizing the high-order derivatives on grid points near the boundary.It is very challenging to design numerical methods that can efficiently and accurately handle both difficulties.Applying an implicit scheme may be able to remove the stability constraints on the time step,however,it usually requires solving a large global system of nonlinear equations for each time step,and the computational cost could be significant.Integration factor(IF)or exponential time differencing(ETD)methods are one of the popular methods for temporal partial differential equations(PDEs)among many other methods.In our paper,we couple ETD methods with an embedded boundary method to solve a system of reaction-diffusion equations with complex geometries.In particular,we rewrite all ETD schemes into a linear combination of specificФ-functions and apply one state-of-the-art algorithm to compute the matrix-vector multiplications,which offers significant computational advantages with adaptive Krylov subspaces.In addition,we extend this method by incorporating the level set method to solve the free boundary problem.The accuracy,stability,and efficiency of the developed method are demonstrated by numerical examples.
文摘A class of singularly perturbed initial boundary value problems for the reaction diffusion equations in a part of domain are considered. Using the operator theory the asymptotic behavior of solution for the problems is studied.
基金the National Natural Science Foundation of China(12061037,11971209)the Natural Science Foundation of Jiangxi Province(20212BAB201016)National Natural Science Foundation of China(11861038)。
文摘This paper is concerned with the asymptotic behavior of solutions to the initial boundary problem of the two-dimensional density-dependent Boussinesq equations.It is shown that the solutions of the Boussinesq equations converge to those of zero thermal diffusivity Boussinesq equations as the thermal diffusivity tends to zero,and the convergence rate is established.In addition,we prove that the boundary-layer thickness is of the valueδ(k)=k^(α)with anyα∈(0,1/4)for a small diffusivity coefficient k>0,and we also find a function to describe the properties of the boundary layer.
基金The Importent Study Profect of the National Natural Science Poundation of China(90211004)The Natural Sciences Foundation of Zheiiang(102009)
文摘A class of nonlinear singularly perturbed problems for reaction diffusion equations are considered. Under suitable conditions, by using the theory of differential inequalities, the asymptotic behavior of solutions for the initial boundary value problems are studied, reduced problems of which possess two intersecting solutions.