This paper presents a method to solve the problems of solutions for integer differential and partial differential equations using the convergence of Adomian's Method. In this paper, we firstly use the convergence ...This paper presents a method to solve the problems of solutions for integer differential and partial differential equations using the convergence of Adomian's Method. In this paper, we firstly use the convergence of Adomian's Method to derive the solutions of high order linear fractional equations, and then the numerical solutions for nonlinear fractional equations. we also get the solutions of two fractional reaction-diffusion equations.We can see the advantage of this method to deal with fractional differential equations.展开更多
We present an efficient and elementary method to find the partial fraction decomposition of a rational function when the denominator is a product of two highly powered linear factors.
In this paper,the Lie symmetry analysis method is applied to the(2+1)-dimensional time-fractional Heisenberg ferromagnetic spin chain equation.We obtain all the Lie symmetries admitted by the governing equation and re...In this paper,the Lie symmetry analysis method is applied to the(2+1)-dimensional time-fractional Heisenberg ferromagnetic spin chain equation.We obtain all the Lie symmetries admitted by the governing equation and reduce the corresponding(2+1)-dimensional fractional partial differential equations with the Riemann–Liouville fractional derivative to(1+1)-dimensional counterparts with the Erdélyi–Kober fractional derivative.Then,we obtain the power series solutions of the reduced equations,prove their convergence and analyze their dynamic behavior graphically.In addition,the conservation laws for all the obtained Lie symmetries are constructed using the new conservation theorem and the generalization of Noether operators.展开更多
In order to constrain temperature during subduction and subsequent exhumation of fel- sic continental crust, we carried out a Ti-in-zircon thermometer coupled with zircon internal structure and U-Pb age on migmatitic ...In order to constrain temperature during subduction and subsequent exhumation of fel- sic continental crust, we carried out a Ti-in-zircon thermometer coupled with zircon internal structure and U-Pb age on migmatitic gneisses from the Weihai region in the Sulu ultra-high pres- sure （UHP） metamorphic terrane, eastern China. The Weihai migmatitic gneisses are composed of in- tercalated compositional layers of melanosome and plagioclase （Pl）-rich lencosome and K-feldspar （Kfs）-rich pegmatite veins. Four stages of zircon growth were recognized in the Weihai migmatitic gneisses. They successively recorded informations of protolith, prograde metamorphism, decompres- sional partial melting during early stage exhumation and subsequent fractional crystallization of pri- mary melt during later stage cooling exhumation. The inherited cores in zircon from the melanosome and the Pl-rich leucosome suggest that the pro- tolith of the migmatitic gneiss is Mid- Neoproterozoic （-780 Ma） magmatic rock. Metamorphic zircons with concordant ages ranging from 243 to 256 Ma occur as over- growth mantles on the protolith magmatic zir- con cores. The estimated growth temperatures （625-717 ＂C） of the metamorphic zircons have a negative correlation with their ages, indicating a progressive metamorphism in HP eciogite-facies condition during subduction. Zircon recrystal- lized rims （228-2 Ma） in the PI-rich ieucosome layers provide the lower limit of the decompress-sional partial melting time during exhumation. The ages from 228^-2 to 219~2 Ma recorded in the Pl-rich leucosome and the Kfs-rich pegmatite vein, respectively, suggest the duration of the fractional crystallization of primary melt during exhumation. The calculated growth temperatures of the zircon rims from the Pl-rich leucosome range from 858 to 739 , and the temperatures of new growth zircon grains （219±2 Ma） in Kfs-rich vein are between 769 and 529 . The estimated temperatures have a positive correlation with ages from the Pl-rich leucosome to the Kfs-rich pegmatite vein, strongly indi- cating that a process of fractional crystallization of the partial melt during exhumation.展开更多
In this paper, the (G′/G)-expansion method is extended to solve fractional partial differential equations in the sense of modified Riemann-Liouville derivative. Based on a nonlinear fractional complex transformation,...In this paper, the (G′/G)-expansion method is extended to solve fractional partial differential equations in the sense of modified Riemann-Liouville derivative. Based on a nonlinear fractional complex transformation, a certain fractional partial differential equation can be turned into another ordinary differential equation of integer order. For illustrating the validity of this method, we apply it to the space-time fractional generalized Hirota-Satsuma coupled KdV equations and the time-fractional fifth-order Sawada-Kotera equation. As a result, some new exact solutions for them are successfully established.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
In this paper, a consistent Riccati expansion method is developed to solve nonlinear fractional partial differential equations involving Jumarie＇s modified Riemann–Liouville derivative. The efficiency and power of t...In this paper, a consistent Riccati expansion method is developed to solve nonlinear fractional partial differential equations involving Jumarie＇s modified Riemann–Liouville derivative. The efficiency and power of this approach are demonstrated by applying it successfully to some important fractional differential equations, namely, the time fractional Burgers, fractional Sawada–Kotera, and fractional coupled mKdV equation. A variety of new exact solutions to these equations under study are constructed.展开更多
In this paper,two classes of Riesz space fractional partial differential equations including space-fractional and space-time-fractional ones are considered.These two models can be regarded as the generalization of the...In this paper,two classes of Riesz space fractional partial differential equations including space-fractional and space-time-fractional ones are considered.These two models can be regarded as the generalization of the classical wave equation in two space dimensions.Combining with the Crank-Nicolson method in temporal direction,efficient alternating direction implicit Galerkin finite element methods for solving these two fractional models are developed,respectively.The corresponding stability and convergence analysis of the numerical methods are discussed.Numerical results are provided to verify the theoretical analysis.展开更多
In this paper, a new fractional projective Riccati equation method is proposed to establish exact solutions for fractional partial differential equations in the sense of modified Riemann–Liouville derivative. This me...In this paper, a new fractional projective Riccati equation method is proposed to establish exact solutions for fractional partial differential equations in the sense of modified Riemann–Liouville derivative. This method can be seen as the fractional version of the known projective Riccati equation method. For illustrating the validity of this method,we apply this method to solve the space-time fractional Whitham–Broer–Kaup(WBK) equations and the nonlinear fractional Sharma–Tasso–Olever(STO) equation, and as a result, some new exact solutions for them are obtained.展开更多
Motivated by the widely used ans¨atz method and starting from the modified Riemann–Liouville derivative together with a fractional complex transformation that can be utilized to transform nonlinear fractional pa...Motivated by the widely used ans¨atz method and starting from the modified Riemann–Liouville derivative together with a fractional complex transformation that can be utilized to transform nonlinear fractional partial differential equations to nonlinear ordinary differential equations, new types of exact traveling wave solutions to three important nonlinear space- and time-fractional partial differential equations are obtained simultaneously in terms of solutions of a Riccati equation. The results are new and first reported in this paper.展开更多
A framework to obtain numerical solution of the fractional partial differential equation using Bernstein polynomials is presented. The main characteristic behind this approach is that a fractional order operational ma...A framework to obtain numerical solution of the fractional partial differential equation using Bernstein polynomials is presented. The main characteristic behind this approach is that a fractional order operational matrix of Bernstein polynomials is derived. With the operational matrix, the equation is transformed into the products of several dependent matrixes which can also be regarded as the system of linear equations after dispersing the variable. By solving the linear equations, the numerical solutions are acquired. Only a small number of Bernstein polynomials are needed to obtain a satisfactory result. Numerical examples are provided to show that the method is computationally efficient.展开更多
The present paper deals with the numerical solution of time-fractional partial differential equations using the element-free Galerkin （EFG） method, which is based on the moving least-square approximation. Compared w...The present paper deals with the numerical solution of time-fractional partial differential equations using the element-free Galerkin （EFG） method, which is based on the moving least-square approximation. Compared with numerical methods based on meshes, the EFG method for time-fractional partial differential equations needs only scattered nodes instead of meshing the domain of the problem. It neither requires element connectivity nor suffers much degradation in accuracy when nodal arrangements are very irregular. In this method, the first-order time derivative is replaced by the Caputo fractional derivative of order α（0 〈 α≤ 1）. The Galerkin weak form is used to obtain the discrete equations, and the essential boundary conditions are enforced by the penalty method. Several numerical examples are presented and the results we obtained are in good agreement with the exact solutions.展开更多
The nonlinearity inmany problems occurs because of the complexity of the given physical phenomena.The present paper investigates the non-linear fractional partial differential equations’solutions using the Caputo ope...The nonlinearity inmany problems occurs because of the complexity of the given physical phenomena.The present paper investigates the non-linear fractional partial differential equations’solutions using the Caputo operator with Laplace residual power seriesmethod.It is found that the present technique has a direct and simple implementation to solve the targeted problems.The comparison of the obtained solutions has been done with actual solutions to the problems.The fractional-order solutions are presented and considered to be the focal point of this research article.The results of the proposed technique are highly accurate and provide useful information about the actual dynamics of each problem.Because of the simple implementation,the present technique can be extended to solve other important fractional order problems.展开更多
It is well-known that using the traditional reproducing kernel method(TRKM) for solving the fractional partial differential equation(FPDE) is very intractable. In this study, the adaptive single piecewise interpolatio...It is well-known that using the traditional reproducing kernel method(TRKM) for solving the fractional partial differential equation(FPDE) is very intractable. In this study, the adaptive single piecewise interpolation reproducing kernel method(ASPIRKM) is determined to solve the FPDE. This improved method not only improves the calculation accuracy, but also reduces the waste of time. Two numerical examples show that the ASPIRKM is a more time-saving numerical method than the TRKM.展开更多
Emblica officinalis （E. oJficinalis） dried fruits were evaluated for its antitrypanosomal activity and cytotoxic effects. Vero cell line maintained in DMEM （Dubecco＇s Modified Eagle Medium） and incubated with Try...Emblica officinalis （E. oJficinalis） dried fruits were evaluated for its antitrypanosomal activity and cytotoxic effects. Vero cell line maintained in DMEM （Dubecco＇s Modified Eagle Medium） and incubated with Trypanosoma evansi for more than 12 h. MPE was added to the Vero cell culture medium at different concentrations （250-1,000 μg/mL） with trypanosomes concentration （1 × 106 trypanosomes/mL in each ELISA plate well） and incubated at appropriate conditions for 72 h. In-vitro cytotoxieity of MPE of E. officinalis was determined on Vero cells at concentrations （（1.56-100 ~tg/mL）. Acute toxicity and in-vivo infectivity tests were done in mice. Obtained MPE ofE. officinalis underwent process of purification via column chromatography, preparative chromatography and HPLC （higher performance liquid chromatography） with bioassay at different strata on Alsever＇s medium. In-vivo assay for trypanocidal activity, MPE and PPFs （partially purified fractions） of E. officinalis with two sets of mice, each mouse was inoculated with 1 × 104/mL oftrypanosomes and treated （48 h post inoculation） at concentrations （12.5, 25, 50, 100 and 200 mg/kg body weight） were administered at dose rate of 100 [tL per mouse via intraperitoneal route （in treating parassitemic mice） to different groups of mice, 6 mice per concentration. HPLC of partially purified fractions ofE. officinalis was carried out with mobile phase ofacetonitdle： water （40：60） in gradient mode. In vitro, MPE induced immobilization and killing of the parasites in concentration-time dependent manner. Significant reduction of trypanosomes counts from concentration of 250μg/mL and complete killing of trypanosomes at 5th hour of observation, which was statistically equivalent to 4th hour of Diminazine Aceturate （Berenil）, standard reference drug used. HPLC of the partially purified fractions revealed two major prominent peaks at retention time of 1-4 min. In vivo, both MPE and PPFs of test material did prolong lives of mice by 6-9 days but could not cure them. At concentration of 2,000 kg/kg body weight of MPE in acute test, all mice survived. For in-vivo infectivity test, mice injected with immobilized trypanosomes developed parasitemia and died while, the other group survived. MPE, PPFs and Diminazine Aceturate were toxic to Vero cells at all concentrations exception of 1.56, 1.56-3.13 and 1.56-6.25 μg/mL, respectively. From this report, PPFs ofE. officinalis dried fruits demonstrated potential pathway for a new development oftrypanocide in near future if additional investigations are put in place.展开更多
Production of cocoa butter replacer (CBR) from tea seed oil through common modification methods of oils (dry fractionation, partial hydrogenation, chemical and enzymatic interesterification) was evaluated. Some physic...Production of cocoa butter replacer (CBR) from tea seed oil through common modification methods of oils (dry fractionation, partial hydrogenation, chemical and enzymatic interesterification) was evaluated. Some physico-chemical properties (iodine, saponification, acid and peroxide values) and fatty acid composition (FAC) of modified samples were analyzed and compared with a reference cocoa butter (CB). Solid and liquid fractions for large amounts of unsaturated fatty acids (approx. 80%) and thereby lower iodine values (81 - 85 gI2/100g) than that of CB (37% and 34 gI2/100 g, respectively), are not suitable as CBR. Among all ratios of chemically and enzaymatically interesterified oil blends (20%, 25% and 30% of hydrogenated tea seed oil with 80%, 75% and 70% of tea seed oil/liquid fraction/solid fraction), the samples with ratio of 30:70 from both chemical and enzymatic interesterification had FAC and iodine value closer to that of CB. A comparision between chemically and enzymatically interesterified samples (CISs and EISs, respectively), in terms of solid fat content (SFC) indicated that although the SFC values in EIS were much lower than that of CB, but the thermal behavior of this sample is comprable to CB at 20℃- 30℃ (sharp melting point of CB).展开更多
In this paper, we will establish the sufficient conditions for the oscillation of solutions of neutral time fractional partial differential equation of the form for where Ω??is a bounded domain in RN with a ...In this paper, we will establish the sufficient conditions for the oscillation of solutions of neutral time fractional partial differential equation of the form for where Ω??is a bounded domain in RN with a piecewise smooth boundary ?is a constant, is the Riemann-Liouville fractional derivative of order a?of u with respect to t and is the Laplacian operator in the Euclidean N-space RN subject to the展开更多
In this paper, we present a new algorithm to solve a kind of nonlinear time space-fractional partial differential equations on a finite domain. The method is based on B-spline wavelets approximations, some of these fu...In this paper, we present a new algorithm to solve a kind of nonlinear time space-fractional partial differential equations on a finite domain. The method is based on B-spline wavelets approximations, some of these functions are reshaped to satisfy on boundary conditions exactly. The Adams fractional method is used to reduce the problem to a system of equations. By multiscale method this system is divided into some smaller systems which have less computations. We get an approximated solution which is more accurate on some subdomains by combining the solutions of these systems. Illustrative examples are included to demonstrate the validity and applicability of our proposed technique, also the stability of the method is discussed.展开更多
The Cauchy problem for some parabolic fractional partial differential equation of higher orders and with time delays is considered. The existence and unique solution of this problem is studied. Some smoothness propert...The Cauchy problem for some parabolic fractional partial differential equation of higher orders and with time delays is considered. The existence and unique solution of this problem is studied. Some smoothness properties with respect to the parameters of these delay fractional differential equations are considered.展开更多
A stochastic averaging method for predicting the response of quasi partially integrable and non-resonant Hamiltoniansystems to fractional Gaussian noise （fGla） with the Hurst index 1/2〈H〈l is proposed. The average...A stochastic averaging method for predicting the response of quasi partially integrable and non-resonant Hamiltoniansystems to fractional Gaussian noise （fGla） with the Hurst index 1/2〈H〈l is proposed. The averaged stochastic differential equa-tions （SDEs） for the first integrals of the associated Hamiltonian system are derived. The dimension of averaged SDEs is less thanthat of the original system. The stationary probability density and statistics of the original system are obtained approximately fromsolving the averaged SDEs numerically. Two systems are worked out to illustrate the proposed stochastic averaging method. It isshown that the results obtained by using the proposed stochastic averaging method and those from digital simulation of originalsystem agree well, and the computational time for the former results is less than that for the latter ones.展开更多
基金the National Natural Science Foundation of China(Nos.11601003,11371027)Natural Science Research Project of Colleges of Anhui Province(No.KJ2016A023)+1 种基金Natural Science Foundation of Anhui Province(No.1508085MA01)College Students’Scientific Research Training Plan of Anhui University(No.KYXL2014006)
文摘This paper presents a method to solve the problems of solutions for integer differential and partial differential equations using the convergence of Adomian's Method. In this paper, we firstly use the convergence of Adomian's Method to derive the solutions of high order linear fractional equations, and then the numerical solutions for nonlinear fractional equations. we also get the solutions of two fractional reaction-diffusion equations.We can see the advantage of this method to deal with fractional differential equations.
文摘We present an efficient and elementary method to find the partial fraction decomposition of a rational function when the denominator is a product of two highly powered linear factors.
基金supported by the State Key Program of the National Natural Science Foundation of China(72031009).
文摘In this paper,the Lie symmetry analysis method is applied to the(2+1)-dimensional time-fractional Heisenberg ferromagnetic spin chain equation.We obtain all the Lie symmetries admitted by the governing equation and reduce the corresponding(2+1)-dimensional fractional partial differential equations with the Riemann–Liouville fractional derivative to(1+1)-dimensional counterparts with the Erdélyi–Kober fractional derivative.Then,we obtain the power series solutions of the reduced equations,prove their convergence and analyze their dynamic behavior graphically.In addition,the conservation laws for all the obtained Lie symmetries are constructed using the new conservation theorem and the generalization of Noether operators.
基金supported by the National Key Basic Research Program of China (No.2009CB825001)the National Natural Science Foundation of China (Nos.40603002,41072046,and 41090371)the Fundamental Research Funds for the Central Universities,China University of Geosciences,Wuhan (No.CUG120121)
文摘In order to constrain temperature during subduction and subsequent exhumation of fel- sic continental crust, we carried out a Ti-in-zircon thermometer coupled with zircon internal structure and U-Pb age on migmatitic gneisses from the Weihai region in the Sulu ultra-high pres- sure （UHP） metamorphic terrane, eastern China. The Weihai migmatitic gneisses are composed of in- tercalated compositional layers of melanosome and plagioclase （Pl）-rich lencosome and K-feldspar （Kfs）-rich pegmatite veins. Four stages of zircon growth were recognized in the Weihai migmatitic gneisses. They successively recorded informations of protolith, prograde metamorphism, decompres- sional partial melting during early stage exhumation and subsequent fractional crystallization of pri- mary melt during later stage cooling exhumation. The inherited cores in zircon from the melanosome and the Pl-rich leucosome suggest that the pro- tolith of the migmatitic gneiss is Mid- Neoproterozoic （-780 Ma） magmatic rock. Metamorphic zircons with concordant ages ranging from 243 to 256 Ma occur as over- growth mantles on the protolith magmatic zir- con cores. The estimated growth temperatures （625-717 ＂C） of the metamorphic zircons have a negative correlation with their ages, indicating a progressive metamorphism in HP eciogite-facies condition during subduction. Zircon recrystal- lized rims （228-2 Ma） in the PI-rich ieucosome layers provide the lower limit of the decompress-sional partial melting time during exhumation. The ages from 228^-2 to 219~2 Ma recorded in the Pl-rich leucosome and the Kfs-rich pegmatite vein, respectively, suggest the duration of the fractional crystallization of primary melt during exhumation. The calculated growth temperatures of the zircon rims from the Pl-rich leucosome range from 858 to 739 , and the temperatures of new growth zircon grains （219±2 Ma） in Kfs-rich vein are between 769 and 529 . The estimated temperatures have a positive correlation with ages from the Pl-rich leucosome to the Kfs-rich pegmatite vein, strongly indi- cating that a process of fractional crystallization of the partial melt during exhumation.
文摘In this paper, the (G′/G)-expansion method is extended to solve fractional partial differential equations in the sense of modified Riemann-Liouville derivative. Based on a nonlinear fractional complex transformation, a certain fractional partial differential equation can be turned into another ordinary differential equation of integer order. For illustrating the validity of this method, we apply it to the space-time fractional generalized Hirota-Satsuma coupled KdV equations and the time-fractional fifth-order Sawada-Kotera equation. As a result, some new exact solutions for them are successfully established.
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.
基金Supported by the National Natural Science Foundation of China under Grant Nos.11101332,11201371,11371293 the Natural Science Foundation of Shaanxi Province under Grant No.2015JM1037
文摘In this paper, a consistent Riccati expansion method is developed to solve nonlinear fractional partial differential equations involving Jumarie＇s modified Riemann–Liouville derivative. The efficiency and power of this approach are demonstrated by applying it successfully to some important fractional differential equations, namely, the time fractional Burgers, fractional Sawada–Kotera, and fractional coupled mKdV equation. A variety of new exact solutions to these equations under study are constructed.
基金supported by the Guangxi Natural Science Foundation[grant numbers 2018GXNSFBA281020,2018GXNSFAA138121]the Doctoral Starting up Foundation of Guilin University of Technology[grant number GLUTQD2016044].
文摘In this paper,two classes of Riesz space fractional partial differential equations including space-fractional and space-time-fractional ones are considered.These two models can be regarded as the generalization of the classical wave equation in two space dimensions.Combining with the Crank-Nicolson method in temporal direction,efficient alternating direction implicit Galerkin finite element methods for solving these two fractional models are developed,respectively.The corresponding stability and convergence analysis of the numerical methods are discussed.Numerical results are provided to verify the theoretical analysis.
基金Supported by Natural Science Foundation of Shandong Province of China under Grant No.ZR2013AQ009National Training Programs of Innovation and Entrepreneurship for Undergraduates under Grant No.201310433031Doctoral initializing Foundation of Shandong University of Technology of China under Grant No.4041-413030
文摘In this paper, a new fractional projective Riccati equation method is proposed to establish exact solutions for fractional partial differential equations in the sense of modified Riemann–Liouville derivative. This method can be seen as the fractional version of the known projective Riccati equation method. For illustrating the validity of this method,we apply this method to solve the space-time fractional Whitham–Broer–Kaup(WBK) equations and the nonlinear fractional Sharma–Tasso–Olever(STO) equation, and as a result, some new exact solutions for them are obtained.
基金Supported by National Natural Science Foundation of China under Grant Nos.11071278,111471004the Fundamental Research Funds for the Central Universities of GK201302026 and GK201102007
文摘Motivated by the widely used ans¨atz method and starting from the modified Riemann–Liouville derivative together with a fractional complex transformation that can be utilized to transform nonlinear fractional partial differential equations to nonlinear ordinary differential equations, new types of exact traveling wave solutions to three important nonlinear space- and time-fractional partial differential equations are obtained simultaneously in terms of solutions of a Riccati equation. The results are new and first reported in this paper.
基金supported by the Natural Science Foundation of Hebei Province under Grant No.A2012203407
文摘A framework to obtain numerical solution of the fractional partial differential equation using Bernstein polynomials is presented. The main characteristic behind this approach is that a fractional order operational matrix of Bernstein polynomials is derived. With the operational matrix, the equation is transformed into the products of several dependent matrixes which can also be regarded as the system of linear equations after dispersing the variable. By solving the linear equations, the numerical solutions are acquired. Only a small number of Bernstein polynomials are needed to obtain a satisfactory result. Numerical examples are provided to show that the method is computationally efficient.
基金Project supported by the National Natural Science Foundation of China(Grant No.11072117)the Natural Science Foundationof Zhejiang Province,China(Grant Nos.Y6110007and Y6110502)the K.C.Wong Magna Fund in Ningbo University,China
文摘The present paper deals with the numerical solution of time-fractional partial differential equations using the element-free Galerkin （EFG） method, which is based on the moving least-square approximation. Compared with numerical methods based on meshes, the EFG method for time-fractional partial differential equations needs only scattered nodes instead of meshing the domain of the problem. It neither requires element connectivity nor suffers much degradation in accuracy when nodal arrangements are very irregular. In this method, the first-order time derivative is replaced by the Caputo fractional derivative of order α（0 〈 α≤ 1）. The Galerkin weak form is used to obtain the discrete equations, and the essential boundary conditions are enforced by the penalty method. Several numerical examples are presented and the results we obtained are in good agreement with the exact solutions.
基金Supporting Project No.(RSP-2021/401),King Saud University,Riyadh,Saudi Arabia.
文摘The nonlinearity inmany problems occurs because of the complexity of the given physical phenomena.The present paper investigates the non-linear fractional partial differential equations’solutions using the Caputo operator with Laplace residual power seriesmethod.It is found that the present technique has a direct and simple implementation to solve the targeted problems.The comparison of the obtained solutions has been done with actual solutions to the problems.The fractional-order solutions are presented and considered to be the focal point of this research article.The results of the proposed technique are highly accurate and provide useful information about the actual dynamics of each problem.Because of the simple implementation,the present technique can be extended to solve other important fractional order problems.
基金Natural Science Foundation of Inner Mongolia Autonomous Region of China (No.2019BS01011)Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region,China (No.NJYT-20-B18)2022 Talent Development Foundation of Inner Mongolia Autonomous Region,China。
文摘It is well-known that using the traditional reproducing kernel method(TRKM) for solving the fractional partial differential equation(FPDE) is very intractable. In this study, the adaptive single piecewise interpolation reproducing kernel method(ASPIRKM) is determined to solve the FPDE. This improved method not only improves the calculation accuracy, but also reduces the waste of time. Two numerical examples show that the ASPIRKM is a more time-saving numerical method than the TRKM.
文摘Emblica officinalis （E. oJficinalis） dried fruits were evaluated for its antitrypanosomal activity and cytotoxic effects. Vero cell line maintained in DMEM （Dubecco＇s Modified Eagle Medium） and incubated with Trypanosoma evansi for more than 12 h. MPE was added to the Vero cell culture medium at different concentrations （250-1,000 μg/mL） with trypanosomes concentration （1 × 106 trypanosomes/mL in each ELISA plate well） and incubated at appropriate conditions for 72 h. In-vitro cytotoxieity of MPE of E. officinalis was determined on Vero cells at concentrations （（1.56-100 ~tg/mL）. Acute toxicity and in-vivo infectivity tests were done in mice. Obtained MPE ofE. officinalis underwent process of purification via column chromatography, preparative chromatography and HPLC （higher performance liquid chromatography） with bioassay at different strata on Alsever＇s medium. In-vivo assay for trypanocidal activity, MPE and PPFs （partially purified fractions） of E. officinalis with two sets of mice, each mouse was inoculated with 1 × 104/mL oftrypanosomes and treated （48 h post inoculation） at concentrations （12.5, 25, 50, 100 and 200 mg/kg body weight） were administered at dose rate of 100 [tL per mouse via intraperitoneal route （in treating parassitemic mice） to different groups of mice, 6 mice per concentration. HPLC of partially purified fractions ofE. officinalis was carried out with mobile phase ofacetonitdle： water （40：60） in gradient mode. In vitro, MPE induced immobilization and killing of the parasites in concentration-time dependent manner. Significant reduction of trypanosomes counts from concentration of 250μg/mL and complete killing of trypanosomes at 5th hour of observation, which was statistically equivalent to 4th hour of Diminazine Aceturate （Berenil）, standard reference drug used. HPLC of the partially purified fractions revealed two major prominent peaks at retention time of 1-4 min. In vivo, both MPE and PPFs of test material did prolong lives of mice by 6-9 days but could not cure them. At concentration of 2,000 kg/kg body weight of MPE in acute test, all mice survived. For in-vivo infectivity test, mice injected with immobilized trypanosomes developed parasitemia and died while, the other group survived. MPE, PPFs and Diminazine Aceturate were toxic to Vero cells at all concentrations exception of 1.56, 1.56-3.13 and 1.56-6.25 μg/mL, respectively. From this report, PPFs ofE. officinalis dried fruits demonstrated potential pathway for a new development oftrypanocide in near future if additional investigations are put in place.
文摘Production of cocoa butter replacer (CBR) from tea seed oil through common modification methods of oils (dry fractionation, partial hydrogenation, chemical and enzymatic interesterification) was evaluated. Some physico-chemical properties (iodine, saponification, acid and peroxide values) and fatty acid composition (FAC) of modified samples were analyzed and compared with a reference cocoa butter (CB). Solid and liquid fractions for large amounts of unsaturated fatty acids (approx. 80%) and thereby lower iodine values (81 - 85 gI2/100g) than that of CB (37% and 34 gI2/100 g, respectively), are not suitable as CBR. Among all ratios of chemically and enzaymatically interesterified oil blends (20%, 25% and 30% of hydrogenated tea seed oil with 80%, 75% and 70% of tea seed oil/liquid fraction/solid fraction), the samples with ratio of 30:70 from both chemical and enzymatic interesterification had FAC and iodine value closer to that of CB. A comparision between chemically and enzymatically interesterified samples (CISs and EISs, respectively), in terms of solid fat content (SFC) indicated that although the SFC values in EIS were much lower than that of CB, but the thermal behavior of this sample is comprable to CB at 20℃- 30℃ (sharp melting point of CB).
文摘In this paper, we will establish the sufficient conditions for the oscillation of solutions of neutral time fractional partial differential equation of the form for where Ω??is a bounded domain in RN with a piecewise smooth boundary ?is a constant, is the Riemann-Liouville fractional derivative of order a?of u with respect to t and is the Laplacian operator in the Euclidean N-space RN subject to the
文摘In this paper, we present a new algorithm to solve a kind of nonlinear time space-fractional partial differential equations on a finite domain. The method is based on B-spline wavelets approximations, some of these functions are reshaped to satisfy on boundary conditions exactly. The Adams fractional method is used to reduce the problem to a system of equations. By multiscale method this system is divided into some smaller systems which have less computations. We get an approximated solution which is more accurate on some subdomains by combining the solutions of these systems. Illustrative examples are included to demonstrate the validity and applicability of our proposed technique, also the stability of the method is discussed.
文摘The Cauchy problem for some parabolic fractional partial differential equation of higher orders and with time delays is considered. The existence and unique solution of this problem is studied. Some smoothness properties with respect to the parameters of these delay fractional differential equations are considered.
基金supported by the National Natural Science Foundation of China(Nos.11172259,11272279,11321202,and 11432012)
文摘A stochastic averaging method for predicting the response of quasi partially integrable and non-resonant Hamiltoniansystems to fractional Gaussian noise （fGla） with the Hurst index 1/2〈H〈l is proposed. The averaged stochastic differential equa-tions （SDEs） for the first integrals of the associated Hamiltonian system are derived. The dimension of averaged SDEs is less thanthat of the original system. The stationary probability density and statistics of the original system are obtained approximately fromsolving the averaged SDEs numerically. Two systems are worked out to illustrate the proposed stochastic averaging method. It isshown that the results obtained by using the proposed stochastic averaging method and those from digital simulation of originalsystem agree well, and the computational time for the former results is less than that for the latter ones.
