This paper is devoted to the investigation of the Landau–Ginzburg–Higgs equation(LGHe),which serves as a mathematical model to understand phenomena such as superconductivity and cyclotron waves.The LGHe finds applic...This paper is devoted to the investigation of the Landau–Ginzburg–Higgs equation(LGHe),which serves as a mathematical model to understand phenomena such as superconductivity and cyclotron waves.The LGHe finds applications in various scientific fields,including fluid dynamics,plasma physics,biological systems,and electricity-electronics.The study adopts Lie symmetry analysis as the primary framework for exploration.This analysis involves the identification of Lie point symmetries that are admitted by the differential equation.By leveraging these Lie point symmetries,symmetry reductions are performed,leading to the discovery of group invariant solutions.To obtain explicit solutions,several mathematical methods are applied,including Kudryashov's method,the extended Jacobi elliptic function expansion method,the power series method,and the simplest equation method.These methods yield solutions characterized by exponential,hyperbolic,and elliptic functions.The obtained solutions are visually represented through 3D,2D,and density plots,which effectively illustrate the nature of the solutions.These plots depict various patterns,such as kink-shaped,singular kink-shaped,bell-shaped,and periodic solutions.Finally,the paper employs the multiplier method and the conservation theorem introduced by Ibragimov to derive conserved vectors.These conserved vectors play a crucial role in the study of physical quantities,such as the conservation of energy and momentum,and contribute to the understanding of the underlying physics of the system.展开更多
We prove the existence of solutions of the system for nonlocal Neumann problems with Minkowski-curvature operator(r^(N−1) u′/√(p 1−u′^(2)))′=r^(N−1)f(r,u),r 2(0,1),u′(0)=0,u′(1)=∫_(0)^(-) u′(s)dg(s),where k,N...We prove the existence of solutions of the system for nonlocal Neumann problems with Minkowski-curvature operator(r^(N−1) u′/√(p 1−u′^(2)))′=r^(N−1)f(r,u),r 2(0,1),u′(0)=0,u′(1)=∫_(0)^(-) u′(s)dg(s),where k,N≥1 are integers,f:[0,1]×R^(k)→R^(k) is continuous and g:[0,1]→R^(k) is a function of bounded variation.Our proof is based on the perturbation method.展开更多
In this paper, two kinds of parametric generalized vector quasi-equilibrium problems are introduced and the relations between them are studied. The upper and lower semicontinuity of their solution sets to parameters a...In this paper, two kinds of parametric generalized vector quasi-equilibrium problems are introduced and the relations between them are studied. The upper and lower semicontinuity of their solution sets to parameters are investigated.展开更多
In this article, we study Levitin-Polyak type well-posedness for generalized vector equilibrium problems with abstract and functional constraints. Criteria and characterizations for these types of well-posednesses are...In this article, we study Levitin-Polyak type well-posedness for generalized vector equilibrium problems with abstract and functional constraints. Criteria and characterizations for these types of well-posednesses are given.展开更多
We give an existence result of the obstacle parabolic equations3b（x,u） div（a（x,t,u, Vu））＋div（（x,t,u））=f in QT, 3twhere b（x,u） is bounded function ot u, the term atva,x,r,u, v u）） is a Letay type operat...We give an existence result of the obstacle parabolic equations3b（x,u） div（a（x,t,u, Vu））＋div（（x,t,u））=f in QT, 3twhere b（x,u） is bounded function ot u, the term atva,x,r,u, v u）） is a Letay type operator and the function is a nonlinear lower order and satisfy only the growth condition. The second term belongs to L1 （QT）. The proof of an existence solution is based on the penalization methods.展开更多
In this work,the localized method of fundamental solution(LMFS)is extended to Signorini problem.Unlike the traditional fundamental solution(MFS),the LMFS approximates the field quantity at each node by using the field...In this work,the localized method of fundamental solution(LMFS)is extended to Signorini problem.Unlike the traditional fundamental solution(MFS),the LMFS approximates the field quantity at each node by using the field quantities at the adjacent nodes.The idea of the LMFS is similar to the localized domain type method.The fictitious boundary nodes are proposed to impose the boundary condition and governing equations at each node to formulate a sparse matrix.The inequality boundary condition of Signorini problem is solved indirectly by introducing nonlinear complementarity problem function(NCP-function).Numerical examples are carried out to validate the reliability and effectiveness of the LMFS in solving Signorini problems.展开更多
In this paper, we study a class of boundary value problems for conformable fractional differential equations under a new definition. Firstly, by using the monotone iterative technique and the method of coupled upper a...In this paper, we study a class of boundary value problems for conformable fractional differential equations under a new definition. Firstly, by using the monotone iterative technique and the method of coupled upper and lower solution, the sufficient condition for the existence of the boundary value problem is obtained, and the range of the solution is determined. Then the existence and uniqueness of the solution are proved by the proof by contradiction. Finally, a concrete example is given to illustrate the wide applicability of our main results.展开更多
This paper deals with the forward and backward problems for the nonlinear fractional pseudo-parabolic equation ut+(-Δ)^(s1)ut+β(-Δ)^(s2)u=F(u,x,t)subject o random Gaussian white noise for initial and final data.Und...This paper deals with the forward and backward problems for the nonlinear fractional pseudo-parabolic equation ut+(-Δ)^(s1)ut+β(-Δ)^(s2)u=F(u,x,t)subject o random Gaussian white noise for initial and final data.Under the suitable assumptions s1,s2andβ,we first show the ill-posedness of mild solutions for forward and backward problems in the sense of Hadamard,which are mainly driven by random noise.Moreover,we propose the Fourier truncation method for stabilizing the above ill-posed problems.We derive an error estimate between the exact solution and its regularized solution in an E‖·‖Hs22norm,and give some numerical examples illustrating the effect of above method.展开更多
Novel exact solutions of one-dimensional transient dynamic piezoelectric problems for thickness polarized layers and disks, or length polarized rods, are obtained. The solutions are derived using a time-domain Green’...Novel exact solutions of one-dimensional transient dynamic piezoelectric problems for thickness polarized layers and disks, or length polarized rods, are obtained. The solutions are derived using a time-domain Green’s function method that leads to an exact analytical recursive procedure which is applicable for a wide variety of boundary conditions including nonlinear cases. A nonlinear damper boundary condition is considered in more detail. The corresponding nonlinear relationship between stresses and velocities at a current time moment is used in the recursive procedure. In addition to the exact recursive procedure that is effective for calculations, some new practically important explicit exact solutions are presented. Several examples of the time behavior of the output electric potential difference are given to illustrate the effectiveness of the proposed exact approach.展开更多
We present the optimal homotopy asymptotic method (OHAM) to find the numerical solution of the second order initial value problems of Bratu-type. We solve some examples to illustrate the validity and efficiency of the...We present the optimal homotopy asymptotic method (OHAM) to find the numerical solution of the second order initial value problems of Bratu-type. We solve some examples to illustrate the validity and efficiency of the method.展开更多
In this paper the method of reciprocal theorem is extended to find solutions of plane problems of elasticity of the rectangular plates with various edge conditions.First we give the basic solution of the plane problem...In this paper the method of reciprocal theorem is extended to find solutions of plane problems of elasticity of the rectangular plates with various edge conditions.First we give the basic solution of the plane problem of the rectangular plate with four edges built-in as the basic system and then find displacement expressions of the actual system by using the reciprocal theorem between the basic system and actual system with various edge conditions.When only displacement edge conditions exist, obtaining displacement expressions by means of the method of reciprocal theorem is actual. But in other conditions, when static force edge conditions or mixed ones exist, the obtained displacements are admissible. In order to find actual displacement, the minimum potential energy theorem must be applied.Calculations show that the method of reciprocal theorem is a simple, convenient and general one for the solution of plane problems of elasticity of the rectangular plates with various edge conditions. Evidently, it is a new method.展开更多
We presented Mathematical apparatus of the choice of optimum parameters of technical, technological systems and materials on the basis of vector optimization. We have considered the formulation and solution of three t...We presented Mathematical apparatus of the choice of optimum parameters of technical, technological systems and materials on the basis of vector optimization. We have considered the formulation and solution of three types of tasks presented below. First, the problem of selecting the optimal parameters of technical systems depending on the functional characteristics of the system. Secondly, the problem of selecting the optimal parameters of the process depending on the technological characteristics of the process. Third, the problem of choosing the optimal structure of the material depending on the functional characteristics of this material. The statement of all problems is made in the form of vector problems of mathematical (nonlinear) programming. The theory and the principle of optimality of the solution of vector tasks it is explained in work of https://rdcu.be/bhZ8i. The implementation of the methodology is shown on a numerical example of the choice of optimum parameters of the technical, technological systems and materials. On the basis of mathematical methods of solution of vector problems we developed the software in the MATLAB system. The numerical example includes: input data (requirement specification) for modeling;transformation of mathematical models with uncertainty to the model under certainty;acceptance of an optimal solution with equivalent criteria (the solution of numerical model);acceptance of an optimal solution with the given priority of criterion.展开更多
In this article, we are concerned with the stability of stationary solution for outflow problem on the Navier-Stokes-Poisson system. We obtain the unique existence and the asymptotic stability of stationary solution. ...In this article, we are concerned with the stability of stationary solution for outflow problem on the Navier-Stokes-Poisson system. We obtain the unique existence and the asymptotic stability of stationary solution. Moreover, the convergence rate of solution towards stationary solution is obtained. Precisely, if an initial perturbation decays with the algebraic or the exponential rate in space, the solution converges to the corresponding stationary solution as time tends to infinity with the algebraic or the exponential rate in time. The proof is based on the weighted energy method by taking into account the effect of the self-consistent electric field on the viscous compressible fluid.展开更多
The existence of nondecreasing positive solutions for the nonlinear third-order twopoint boundary value problem u′″（t） ＋ q（t）f（t,u（t）,u′（t）） = 0, 0 〈 t 〈 1, u（0） = u″（0） = u′（1） = 0 is studied....The existence of nondecreasing positive solutions for the nonlinear third-order twopoint boundary value problem u′″（t） ＋ q（t）f（t,u（t）,u′（t）） = 0, 0 〈 t 〈 1, u（0） = u″（0） = u′（1） = 0 is studied. The iterative schemes for approximating the solutions are obtained by applying a monotone iterative method.展开更多
The optimization problem is considered in which the objective function is pseudolinear（both pseudoconvex and pseudoconcave） and the constraints are linear. The general expression for the optimal solutions to the pro...The optimization problem is considered in which the objective function is pseudolinear（both pseudoconvex and pseudoconcave） and the constraints are linear. The general expression for the optimal solutions to the problem is derived with the representation theorem of polyhedral sets, and the uniqueness condition of the optimal solution and the computational procedures to determine all optimal solutions （if the uniqueness condition is not satisfied ） are provided. Finally, an illustrative example is also given.展开更多
The paper deals with a development of the discrete-analytical method for the solution of the dynamical problems of a hollow sphere with inhomogeneous initial stresses.The examinations are made with respect to the prob...The paper deals with a development of the discrete-analytical method for the solution of the dynamical problems of a hollow sphere with inhomogeneous initial stresses.The examinations are made with respect to the problem on the natural vibration of the hollow sphere the initial stresses in which is caused by internal and external uniformly distributed pressure.The initial stresses in the sphere are determined within the scope of the exact equations of elastostatics.It is assumed that after appearing this static initial stresses the sphere gets a dynamical excitation and mechanical behavior of the sphere caused by this excitation is described with the so-called three-dimensional linearized equations of elastic wave propagation in initially stressed bodies.For the solution of these equations,which have variable coefficients,the discrete analytical solution method is developed and applied.In particular,it is established that the convergence of the numerical results with respect to the number of discretization is very acceptable and applicable for the considered type dynamical problems.Numerical results on the influence of the initial stresses on the values of the natural frequencies of the hollow sphere are also presented and these results are discussed.展开更多
A new algorithm is presented for solving Troesch’s problem. The numerical scheme based on the sinc-collocation technique is deduced. The equation is reduced to systems of nonlinear algebraic equations. Some numerical...A new algorithm is presented for solving Troesch’s problem. The numerical scheme based on the sinc-collocation technique is deduced. The equation is reduced to systems of nonlinear algebraic equations. Some numerical experiments are made. Compared with the modified homotopy perturbation technique (MHP), the variational iteration method and the Adomian decomposition method. It is shown that the sinc-collocation method yields better results.展开更多
The purpose of this paper is to investigate a nonlocal Dirichlet problem with(p(x),q(x))-Laplacian-like operator originated from a capillary phenomena.Using the variational methods and the critical point theory,we est...The purpose of this paper is to investigate a nonlocal Dirichlet problem with(p(x),q(x))-Laplacian-like operator originated from a capillary phenomena.Using the variational methods and the critical point theory,we establish the existence of infinitely many weak solutions for this problem.展开更多
Cone-convex, cone-monotonic and positively continuous homogeneous operators are used as duality variables and the Lagrange duality of vector maximization problem in Banach space is discussed. The results are the exten...Cone-convex, cone-monotonic and positively continuous homogeneous operators are used as duality variables and the Lagrange duality of vector maximization problem in Banach space is discussed. The results are the extension of Ref[1,3,4] to some extent.The only tool used in the proof of theorem is Eidelheit separated theorem of two convex sets.展开更多
In this paper, the problem of reconstructing numerical derivatives from noisy data is considered. A new framework of mollification methods based on the L generalized solution regularization methods is proposed. A spec...In this paper, the problem of reconstructing numerical derivatives from noisy data is considered. A new framework of mollification methods based on the L generalized solution regularization methods is proposed. A specific algorithm for the first three derivatives is presented in the paper, in which a modification of TSVD, termed cTSVD is chosen as the regularization technique. Numerical examples given in the paper verify the theoretical results and show efficiency of the new method.展开更多
基金the South African National Space Agency (SANSA) for funding this work
文摘This paper is devoted to the investigation of the Landau–Ginzburg–Higgs equation(LGHe),which serves as a mathematical model to understand phenomena such as superconductivity and cyclotron waves.The LGHe finds applications in various scientific fields,including fluid dynamics,plasma physics,biological systems,and electricity-electronics.The study adopts Lie symmetry analysis as the primary framework for exploration.This analysis involves the identification of Lie point symmetries that are admitted by the differential equation.By leveraging these Lie point symmetries,symmetry reductions are performed,leading to the discovery of group invariant solutions.To obtain explicit solutions,several mathematical methods are applied,including Kudryashov's method,the extended Jacobi elliptic function expansion method,the power series method,and the simplest equation method.These methods yield solutions characterized by exponential,hyperbolic,and elliptic functions.The obtained solutions are visually represented through 3D,2D,and density plots,which effectively illustrate the nature of the solutions.These plots depict various patterns,such as kink-shaped,singular kink-shaped,bell-shaped,and periodic solutions.Finally,the paper employs the multiplier method and the conservation theorem introduced by Ibragimov to derive conserved vectors.These conserved vectors play a crucial role in the study of physical quantities,such as the conservation of energy and momentum,and contribute to the understanding of the underlying physics of the system.
基金Supported by the National Natural Science Foundation of China(Grant Nos.11901464,12361040)the National Science Foundation of Gansu Province(Grant Nos.20JR10RA100,21JR1RA230)the Department of Education University Innovation Fund of Gansu Province(Grant Nos.2022A-218,2021A-006)。
文摘We prove the existence of solutions of the system for nonlocal Neumann problems with Minkowski-curvature operator(r^(N−1) u′/√(p 1−u′^(2)))′=r^(N−1)f(r,u),r 2(0,1),u′(0)=0,u′(1)=∫_(0)^(-) u′(s)dg(s),where k,N≥1 are integers,f:[0,1]×R^(k)→R^(k) is continuous and g:[0,1]→R^(k) is a function of bounded variation.Our proof is based on the perturbation method.
基金The NSF(10871226) of Chinathe NSF(ZR2009AL006) of Shandong Province
文摘In this paper, two kinds of parametric generalized vector quasi-equilibrium problems are introduced and the relations between them are studied. The upper and lower semicontinuity of their solution sets to parameters are investigated.
基金supported by the National Science Foundation of China and Shanghai Pujian Program
文摘In this article, we study Levitin-Polyak type well-posedness for generalized vector equilibrium problems with abstract and functional constraints. Criteria and characterizations for these types of well-posednesses are given.
文摘We give an existence result of the obstacle parabolic equations3b（x,u） div（a（x,t,u, Vu））＋div（（x,t,u））=f in QT, 3twhere b（x,u） is bounded function ot u, the term atva,x,r,u, v u）） is a Letay type operator and the function is a nonlinear lower order and satisfy only the growth condition. The second term belongs to L1 （QT）. The proof of an existence solution is based on the penalization methods.
基金supported by the National Science Foundation of China(No.52109089)support of Post Doctor Program(2019M652281)Nature Science Foundation of Jiangxi Province(20192BAB216040).
文摘In this work,the localized method of fundamental solution(LMFS)is extended to Signorini problem.Unlike the traditional fundamental solution(MFS),the LMFS approximates the field quantity at each node by using the field quantities at the adjacent nodes.The idea of the LMFS is similar to the localized domain type method.The fictitious boundary nodes are proposed to impose the boundary condition and governing equations at each node to formulate a sparse matrix.The inequality boundary condition of Signorini problem is solved indirectly by introducing nonlinear complementarity problem function(NCP-function).Numerical examples are carried out to validate the reliability and effectiveness of the LMFS in solving Signorini problems.
文摘In this paper, we study a class of boundary value problems for conformable fractional differential equations under a new definition. Firstly, by using the monotone iterative technique and the method of coupled upper and lower solution, the sufficient condition for the existence of the boundary value problem is obtained, and the range of the solution is determined. Then the existence and uniqueness of the solution are proved by the proof by contradiction. Finally, a concrete example is given to illustrate the wide applicability of our main results.
基金supported by the Natural Science Foundation of China(11801108)the Natural Science Foundation of Guangdong Province(2021A1515010314)the Science and Technology Planning Project of Guangzhou City(202201010111)。
文摘This paper deals with the forward and backward problems for the nonlinear fractional pseudo-parabolic equation ut+(-Δ)^(s1)ut+β(-Δ)^(s2)u=F(u,x,t)subject o random Gaussian white noise for initial and final data.Under the suitable assumptions s1,s2andβ,we first show the ill-posedness of mild solutions for forward and backward problems in the sense of Hadamard,which are mainly driven by random noise.Moreover,we propose the Fourier truncation method for stabilizing the above ill-posed problems.We derive an error estimate between the exact solution and its regularized solution in an E‖·‖Hs22norm,and give some numerical examples illustrating the effect of above method.
文摘Novel exact solutions of one-dimensional transient dynamic piezoelectric problems for thickness polarized layers and disks, or length polarized rods, are obtained. The solutions are derived using a time-domain Green’s function method that leads to an exact analytical recursive procedure which is applicable for a wide variety of boundary conditions including nonlinear cases. A nonlinear damper boundary condition is considered in more detail. The corresponding nonlinear relationship between stresses and velocities at a current time moment is used in the recursive procedure. In addition to the exact recursive procedure that is effective for calculations, some new practically important explicit exact solutions are presented. Several examples of the time behavior of the output electric potential difference are given to illustrate the effectiveness of the proposed exact approach.
文摘We present the optimal homotopy asymptotic method (OHAM) to find the numerical solution of the second order initial value problems of Bratu-type. We solve some examples to illustrate the validity and efficiency of the method.
文摘In this paper the method of reciprocal theorem is extended to find solutions of plane problems of elasticity of the rectangular plates with various edge conditions.First we give the basic solution of the plane problem of the rectangular plate with four edges built-in as the basic system and then find displacement expressions of the actual system by using the reciprocal theorem between the basic system and actual system with various edge conditions.When only displacement edge conditions exist, obtaining displacement expressions by means of the method of reciprocal theorem is actual. But in other conditions, when static force edge conditions or mixed ones exist, the obtained displacements are admissible. In order to find actual displacement, the minimum potential energy theorem must be applied.Calculations show that the method of reciprocal theorem is a simple, convenient and general one for the solution of plane problems of elasticity of the rectangular plates with various edge conditions. Evidently, it is a new method.
文摘We presented Mathematical apparatus of the choice of optimum parameters of technical, technological systems and materials on the basis of vector optimization. We have considered the formulation and solution of three types of tasks presented below. First, the problem of selecting the optimal parameters of technical systems depending on the functional characteristics of the system. Secondly, the problem of selecting the optimal parameters of the process depending on the technological characteristics of the process. Third, the problem of choosing the optimal structure of the material depending on the functional characteristics of this material. The statement of all problems is made in the form of vector problems of mathematical (nonlinear) programming. The theory and the principle of optimality of the solution of vector tasks it is explained in work of https://rdcu.be/bhZ8i. The implementation of the methodology is shown on a numerical example of the choice of optimum parameters of the technical, technological systems and materials. On the basis of mathematical methods of solution of vector problems we developed the software in the MATLAB system. The numerical example includes: input data (requirement specification) for modeling;transformation of mathematical models with uncertainty to the model under certainty;acceptance of an optimal solution with equivalent criteria (the solution of numerical model);acceptance of an optimal solution with the given priority of criterion.
基金supported by the National Natural Science Foundation of China(11331005,11471134)the Program for Changjiang Scholars and Innovative Research Team in University(IRT13066)the Scientific Research Funds of Huaqiao University(15BS201,15BS309)
文摘In this article, we are concerned with the stability of stationary solution for outflow problem on the Navier-Stokes-Poisson system. We obtain the unique existence and the asymptotic stability of stationary solution. Moreover, the convergence rate of solution towards stationary solution is obtained. Precisely, if an initial perturbation decays with the algebraic or the exponential rate in space, the solution converges to the corresponding stationary solution as time tends to infinity with the algebraic or the exponential rate in time. The proof is based on the weighted energy method by taking into account the effect of the self-consistent electric field on the viscous compressible fluid.
基金Supported by the Natural Science Foundation of Zhejiang Province (Y605144)the XNF of Zhejiang University of Media and Communications (XN080012008034)
文摘The existence of nondecreasing positive solutions for the nonlinear third-order twopoint boundary value problem u′″（t） ＋ q（t）f（t,u（t）,u′（t）） = 0, 0 〈 t 〈 1, u（0） = u″（0） = u′（1） = 0 is studied. The iterative schemes for approximating the solutions are obtained by applying a monotone iterative method.
文摘The optimization problem is considered in which the objective function is pseudolinear（both pseudoconvex and pseudoconcave） and the constraints are linear. The general expression for the optimal solutions to the problem is derived with the representation theorem of polyhedral sets, and the uniqueness condition of the optimal solution and the computational procedures to determine all optimal solutions （if the uniqueness condition is not satisfied ） are provided. Finally, an illustrative example is also given.
文摘The paper deals with a development of the discrete-analytical method for the solution of the dynamical problems of a hollow sphere with inhomogeneous initial stresses.The examinations are made with respect to the problem on the natural vibration of the hollow sphere the initial stresses in which is caused by internal and external uniformly distributed pressure.The initial stresses in the sphere are determined within the scope of the exact equations of elastostatics.It is assumed that after appearing this static initial stresses the sphere gets a dynamical excitation and mechanical behavior of the sphere caused by this excitation is described with the so-called three-dimensional linearized equations of elastic wave propagation in initially stressed bodies.For the solution of these equations,which have variable coefficients,the discrete analytical solution method is developed and applied.In particular,it is established that the convergence of the numerical results with respect to the number of discretization is very acceptable and applicable for the considered type dynamical problems.Numerical results on the influence of the initial stresses on the values of the natural frequencies of the hollow sphere are also presented and these results are discussed.
文摘A new algorithm is presented for solving Troesch’s problem. The numerical scheme based on the sinc-collocation technique is deduced. The equation is reduced to systems of nonlinear algebraic equations. Some numerical experiments are made. Compared with the modified homotopy perturbation technique (MHP), the variational iteration method and the Adomian decomposition method. It is shown that the sinc-collocation method yields better results.
基金Supported by the Fundamental Research Funds for the Central Universities(Grant No.31920180041)the Fund for Talent Introduction of Northwest Minzu University(Grant No.xbmuyjrc201907)。
文摘The purpose of this paper is to investigate a nonlocal Dirichlet problem with(p(x),q(x))-Laplacian-like operator originated from a capillary phenomena.Using the variational methods and the critical point theory,we establish the existence of infinitely many weak solutions for this problem.
文摘Cone-convex, cone-monotonic and positively continuous homogeneous operators are used as duality variables and the Lagrange duality of vector maximization problem in Banach space is discussed. The results are the extension of Ref[1,3,4] to some extent.The only tool used in the proof of theorem is Eidelheit separated theorem of two convex sets.
文摘In this paper, the problem of reconstructing numerical derivatives from noisy data is considered. A new framework of mollification methods based on the L generalized solution regularization methods is proposed. A specific algorithm for the first three derivatives is presented in the paper, in which a modification of TSVD, termed cTSVD is chosen as the regularization technique. Numerical examples given in the paper verify the theoretical results and show efficiency of the new method.
