In this paper, we use the abstract theory of semilinear parabolic equations and a priori estimate techniques to prove the global existence and uniqueness of smooth solutions to the Cauchy problem for the following sys...In this paper, we use the abstract theory of semilinear parabolic equations and a priori estimate techniques to prove the global existence and uniqueness of smooth solutions to the Cauchy problem for the following system of parabolic equations. ψ<sub>t</sub>=-（σ-α）ψ-σθ<sub>x</sub>-αψ<sub>xx</sub> θ<sub>t</sub>=-（1-β）θ-vψ<sub>x</sub>（ψθ）-βθ<sub>xx</sub>展开更多
For the three-dimensional coupled system of multilayer dynamics of fluids in porous media, the second-order upwind finite difference fractional steps schemes applicable to parallel arithmetic are put forward. Some tec...For the three-dimensional coupled system of multilayer dynamics of fluids in porous media, the second-order upwind finite difference fractional steps schemes applicable to parallel arithmetic are put forward. Some techniques, such as calculus of variations, energy method,multiplicative commutation rule of difference operators, decomposition of high order difference operators and prior estimates are adopted. Optimal order estimates in l2 norm are derived to determine the error in the second-order approximate solution. These methods have already been applied to the numerical simulation of migration-accumulation of oil resources.展开更多
It is intended to find the best representation of high-dimensional functions or multivariate data in L2(W) with fewest number of terms, each of them is a combination of one-variable function. A system of nonlinear int...It is intended to find the best representation of high-dimensional functions or multivariate data in L2(W) with fewest number of terms, each of them is a combination of one-variable function. A system of nonlinear integral equations has been derived as an eigenvalue problem of gradient operator in the said space. It proved that the complete set of eigenfunctions generated by the gradient operator constitutes anorthonormal system, and any function of L2(W) can beexpanded with fewest terms and exponential rapidity of convergence. It is also proved as a corollary, thegreatest eigenvalue of the integral operators hasmultiplicity 1 if the dimension of the underlying space nn = 2, 4 and 6.展开更多
The main goal of this work is to develop an effective technique for solving nonlinear systems of Volterra integral equations. The main tools are the cardinal spline functions on small compact supports. We solve a syst...The main goal of this work is to develop an effective technique for solving nonlinear systems of Volterra integral equations. The main tools are the cardinal spline functions on small compact supports. We solve a system of algebra equations to approximate the solution of the system of integral equations. Since the matrix for the algebraic system is nearly triangular, It is relatively painless to solve for the unknowns and an approximation of the original solution with high precision is accomplished. In order to enhance the accuracy, several cardinal splines are employed in the paper. Our schemes were compared with other techniques proposed in recent papers and the advantage of our method was exhibited with several numerical examples.展开更多
Mathematical simulation of nonlinear physical and abstract systems is a very vital process for predicting the solution behavior of fractional partial differential equations(FPDEs)corresponding to different application...Mathematical simulation of nonlinear physical and abstract systems is a very vital process for predicting the solution behavior of fractional partial differential equations(FPDEs)corresponding to different applications in science and engineering. In this paper, an attractive reliable analytical technique, the conformable residual power series, is implemented for constructing approximate series solutions for a class of nonlinear coupled FPDEs arising in fluid mechanics and fluid flow, which are often designed to demonstrate the behavior of weakly nonlinear and long waves and describe the interaction of shallow water waves. In the proposed technique the n-truncated representation is substituted into the original system and it is assumed the(n-1) conformable derivative of the residuum is zero. This allows us to estimate coefficients of truncation and successively add the subordinate terms in the multiple fractional power series with a rapidly convergent form. The influence, capacity, and feasibility of the presented approach are verified by testing some real-world applications. Finally, highlights and some closing comments are attached.展开更多
This research extensively evaluates three leading mathematical software packages: Python, MATLAB, and Scilab, in the context of solving nonlinear systems of equations with five unknown variables. The study’s core obj...This research extensively evaluates three leading mathematical software packages: Python, MATLAB, and Scilab, in the context of solving nonlinear systems of equations with five unknown variables. The study’s core objectives include comparing software performance using standardized benchmarks, employing key performance metrics for quantitative assessment, and examining the influence of varying hardware specifications on software efficiency across HP ProBook, HP EliteBook, Dell Inspiron, and Dell Latitude laptops. Results from this investigation reveal insights into the capabilities of these software tools in diverse computing environments. On the HP ProBook, Python consistently outperforms MATLAB in terms of computational time. Python also exhibits a lower robustness index for problems 3 and 5 but matches or surpasses MATLAB for problem 1, for some initial guess values. In contrast, on the HP EliteBook, MATLAB consistently exhibits shorter computational times than Python across all benchmark problems. However, Python maintains a lower robustness index for most problems, except for problem 3, where MATLAB performs better. A notable challenge is Python’s failure to converge for problem 4 with certain initial guess values, while MATLAB succeeds in producing results. Analysis on the Dell Inspiron reveals a split in strengths. Python demonstrates superior computational efficiency for some problems, while MATLAB excels in handling others. This pattern extends to the robustness index, with Python showing lower values for some problems, and MATLAB achieving the lowest indices for other problems. In conclusion, this research offers valuable insights into the comparative performance of Python, MATLAB, and Scilab in solving nonlinear systems of equations. It underscores the importance of considering both software and hardware specifications in real-world applications. The choice between Python and MATLAB can yield distinct advantages depending on the specific problem and computational environment, providing guidance for 展开更多
文摘In this paper, we use the abstract theory of semilinear parabolic equations and a priori estimate techniques to prove the global existence and uniqueness of smooth solutions to the Cauchy problem for the following system of parabolic equations. ψ<sub>t</sub>=-（σ-α）ψ-σθ<sub>x</sub>-αψ<sub>xx</sub> θ<sub>t</sub>=-（1-β）θ-vψ<sub>x</sub>（ψθ）-βθ<sub>xx</sub>
基金supported by the Major State Basic Research Program of China(Grant No.19990328)the National Tackling Key Problems Program,the National Natural Science Foundation of China(Grant Nos.10372052&10271066)the Doctorate Foundation of the State Education Commission(Grant No.20030422047).
文摘For the three-dimensional coupled system of multilayer dynamics of fluids in porous media, the second-order upwind finite difference fractional steps schemes applicable to parallel arithmetic are put forward. Some techniques, such as calculus of variations, energy method,multiplicative commutation rule of difference operators, decomposition of high order difference operators and prior estimates are adopted. Optimal order estimates in l2 norm are derived to determine the error in the second-order approximate solution. These methods have already been applied to the numerical simulation of migration-accumulation of oil resources.
文摘It is intended to find the best representation of high-dimensional functions or multivariate data in L2(W) with fewest number of terms, each of them is a combination of one-variable function. A system of nonlinear integral equations has been derived as an eigenvalue problem of gradient operator in the said space. It proved that the complete set of eigenfunctions generated by the gradient operator constitutes anorthonormal system, and any function of L2(W) can beexpanded with fewest terms and exponential rapidity of convergence. It is also proved as a corollary, thegreatest eigenvalue of the integral operators hasmultiplicity 1 if the dimension of the underlying space nn = 2, 4 and 6.
文摘The main goal of this work is to develop an effective technique for solving nonlinear systems of Volterra integral equations. The main tools are the cardinal spline functions on small compact supports. We solve a system of algebra equations to approximate the solution of the system of integral equations. Since the matrix for the algebraic system is nearly triangular, It is relatively painless to solve for the unknowns and an approximation of the original solution with high precision is accomplished. In order to enhance the accuracy, several cardinal splines are employed in the paper. Our schemes were compared with other techniques proposed in recent papers and the advantage of our method was exhibited with several numerical examples.
基金Authors gratefully acknowledge Ajman University for providing facilities for our research under Grant Ref.No.2019-IRG-HBS-11.
文摘Mathematical simulation of nonlinear physical and abstract systems is a very vital process for predicting the solution behavior of fractional partial differential equations(FPDEs)corresponding to different applications in science and engineering. In this paper, an attractive reliable analytical technique, the conformable residual power series, is implemented for constructing approximate series solutions for a class of nonlinear coupled FPDEs arising in fluid mechanics and fluid flow, which are often designed to demonstrate the behavior of weakly nonlinear and long waves and describe the interaction of shallow water waves. In the proposed technique the n-truncated representation is substituted into the original system and it is assumed the(n-1) conformable derivative of the residuum is zero. This allows us to estimate coefficients of truncation and successively add the subordinate terms in the multiple fractional power series with a rapidly convergent form. The influence, capacity, and feasibility of the presented approach are verified by testing some real-world applications. Finally, highlights and some closing comments are attached.
文摘This research extensively evaluates three leading mathematical software packages: Python, MATLAB, and Scilab, in the context of solving nonlinear systems of equations with five unknown variables. The study’s core objectives include comparing software performance using standardized benchmarks, employing key performance metrics for quantitative assessment, and examining the influence of varying hardware specifications on software efficiency across HP ProBook, HP EliteBook, Dell Inspiron, and Dell Latitude laptops. Results from this investigation reveal insights into the capabilities of these software tools in diverse computing environments. On the HP ProBook, Python consistently outperforms MATLAB in terms of computational time. Python also exhibits a lower robustness index for problems 3 and 5 but matches or surpasses MATLAB for problem 1, for some initial guess values. In contrast, on the HP EliteBook, MATLAB consistently exhibits shorter computational times than Python across all benchmark problems. However, Python maintains a lower robustness index for most problems, except for problem 3, where MATLAB performs better. A notable challenge is Python’s failure to converge for problem 4 with certain initial guess values, while MATLAB succeeds in producing results. Analysis on the Dell Inspiron reveals a split in strengths. Python demonstrates superior computational efficiency for some problems, while MATLAB excels in handling others. This pattern extends to the robustness index, with Python showing lower values for some problems, and MATLAB achieving the lowest indices for other problems. In conclusion, this research offers valuable insights into the comparative performance of Python, MATLAB, and Scilab in solving nonlinear systems of equations. It underscores the importance of considering both software and hardware specifications in real-world applications. The choice between Python and MATLAB can yield distinct advantages depending on the specific problem and computational environment, providing guidance for
基金supported by the National Natural Science Foundation of China(61370136)the Hainan Province Science and Technology Cooperation Fund Project(KJHZ2015-36)the Hainan Province Introduced and Integrated Demonstration Projects(YJJC20130009)
