In this paper,two types of fractional nonlinear equations in Caputo sense,time-fractional Newell–Whitehead equation(FNWE)and time-fractional generalized Hirota–Satsuma coupled KdV system(HS-cKdVS),are investigated b...In this paper,two types of fractional nonlinear equations in Caputo sense,time-fractional Newell–Whitehead equation(FNWE)and time-fractional generalized Hirota–Satsuma coupled KdV system(HS-cKdVS),are investigated by means of the q-homotopy analysis method(q-HAM).The approximate solutions of the proposed equations are constructed in the form of a convergent series and are compared with the corresponding exact solutions.Due to the presence of the auxiliary parameter h in this method,just a few terms of the series solution are required in order to obtain better approximation.For the sake of visualization,the numerical results obtained in this paper are graphically displayed with the help of Maple.展开更多
This study intends to examine the analytical solutions to the resulting one-dimensional differential equation of acancer tumor model in the frame of time-fractional order with the Caputo-fractional operator employing ...This study intends to examine the analytical solutions to the resulting one-dimensional differential equation of acancer tumor model in the frame of time-fractional order with the Caputo-fractional operator employing a highlyefficient methodology called the q-homotopy analysis transform method.So,the preferred approach effectivelyfound the analytic series solution of the proposed model.The procured outcomes of the present frameworkdemonstrated that this method is authentic for obtaining solutions to a time-fractional-order cancer model.Theresults achieved graphically specify that the concerned paradigm is dependent on arbitrary order and parametersand also disclose the competence of the proposed algorithm.展开更多
In this paper,we analyze the behaviour of solution for the system exemplifying model of tumour invasion and metastasis by the help of q-homotopy analysis transform method(q-HATM)with the fractional operator.The analyz...In this paper,we analyze the behaviour of solution for the system exemplifying model of tumour invasion and metastasis by the help of q-homotopy analysis transform method(q-HATM)with the fractional operator.The analyzed model consists of a system of three nonlinear differential equations elucidating the activation and the migratory response of the degradation of the matrix,tumour cells and production of degradative enzymes by the tumour cells.The considered method is graceful amalgamations of q-homotopy analysis technique with Laplace transform(LT),and Caputo–Fabrizio(CF)fractional operator is hired in the present study.By using the fixed point theory,existence and uniqueness are demonstrated.To validate and present the effectiveness of the considered algorithm,we analyzed the considered system in terms of fractional order with time and space.The error analysis of the considered scheme is illustrated.The variations with small change time with respect to achieved results are effectively captured in plots.The obtained results confirm that the considered method is very efficient and highly methodical to analyze the behaviors of the system of fractional order differential equations.展开更多
In this paper, we consider Variational Iteration Method （VIM） and q-Homotopy Analysis Method （q-HAM） to sotve me partial differential equation resulted from Fingero Imbibition phenomena in double phase flow throug...In this paper, we consider Variational Iteration Method （VIM） and q-Homotopy Analysis Method （q-HAM） to sotve me partial differential equation resulted from Fingero Imbibition phenomena in double phase flow through porous media. We further compare the results obtained here with the solution obtained in [ 12] using Adomian Decomposition Method. Numerical results are obtained, using Mathematica 9, to show the effectiveness of these methods to our choice of problem especially for suitable values ofh and n.展开更多
In this paper, an approximate solution for the one-dimensional hyperbolic telegraph equation by using the q-homotopy analysis method （q-HAM） is proposed.The results shows that the convergence of the q- homotopy anal...In this paper, an approximate solution for the one-dimensional hyperbolic telegraph equation by using the q-homotopy analysis method （q-HAM） is proposed.The results shows that the convergence of the q- homotopy analysis method is more accurate than the convergence of the homotopy analysis method （HAM）.展开更多
This work focuses on designing and analyzing a double parametric fuzzy Homotopy analysis approach with Shehu transform for the non-linear fuzzy time-fractional generalized Fisher’s equation(FTFGFE).A triangular fuzzy...This work focuses on designing and analyzing a double parametric fuzzy Homotopy analysis approach with Shehu transform for the non-linear fuzzy time-fractional generalized Fisher’s equation(FTFGFE).A triangular fuzzy number is used to describe the Caputo fractional derivative(CFD)of order(0,1)that appears in the modeling problem.A novel double parametric form-based Homotopy analysis approach with its convergence analysis is introduced to examine the fuzzy velocities profiles at different spatial positions with crisp and fuzzy conditions.Additional examples are offered to demonstrate the method’s efficacy and viability.The resulting results are compared to otherα=1 results to validate the obtained results and to test the efficiency of the proposed method.The errors approximations are provided to support the suggested computing efficiency of the analytical method.展开更多
In this paper,we find the solutions for fractional potential Korteweg-de Vries(p-KdV)and Benjamin equations using q-homotopy analysis transform method(q-HATM).The considered method is the mixture of q-homotopy analysi...In this paper,we find the solutions for fractional potential Korteweg-de Vries(p-KdV)and Benjamin equations using q-homotopy analysis transform method(q-HATM).The considered method is the mixture of q-homotopy analysis method and Laplace transform,and the Caputo fractional operator is considered in the present investigation.The projected solution procedure manipulates and controls the obtained results in a large admissible domain.Further,it offers a simple algorithm to adjust the convergence province of the obtained solution.To validate the q-HATM is accurate and reliable,the numerical simulations have been conducted for both equations and the outcomes are revealed through the plots and tables.Comparison between the obtained solutions with the exact solutions exhibits that,the considered method is efficient and effective in solving nonlinear problems associated with science and technology.展开更多
The pivotal aim of the present investigation is to find an approximate analytical solution for the system of three fractional differential equations describing the Lakes pollution using q-homotopy analysis transform m...The pivotal aim of the present investigation is to find an approximate analytical solution for the system of three fractional differential equations describing the Lakes pollution using q-homotopy analysis transform method(q-HATM).We consider three different cases of the considered model namely,periodic input model,exponentially decaying input model,and linear input model.The considered scheme is unifications of q-homotopy analysis technique with Laplace transform(LT).To illustrate the existence and uniqueness for the projected model,we consider the fixed point hypothesis.More preciously,we scrutinized the behaviour of the obtained solution for the considered model with fractional-order,in order to elucidate the effectiveness of the proposed algorithm.Further,for the different fractional-order and parameters offered by the considered method,the physical natures have been apprehended.The obtained consequences evidence that the proposed method is very effective and highly methodical to study and examine the nature and its corresponding consequences of the system of fractional order differential equations describing the real word problems.展开更多
The generalized Riccati equation mapping method(GREMM)is used in this paper to obtain different types of soliton solutions for nonlinear Schrödinger equation with higher dimension that existed in the regimes of a...The generalized Riccati equation mapping method(GREMM)is used in this paper to obtain different types of soliton solutions for nonlinear Schrödinger equation with higher dimension that existed in the regimes of anomalous dispersion.Later,we use the q-homotopy analysis method combined with the Laplace transform(q-HATM)to obtain approximate solutions of the bright and dark optical solitons.The q-HATM illustrates the solutions as a rapid convergent series.In addition,to show the physical behavior of the solutions obtained by the proposed techniques,the graphical representation has been provided with some parameter values.The findings demonstrate that the proposed techniques are useful,efficient and reliable mathematical method for the extraction of soliton solutions.展开更多
In the present study,we consider the q-homotopy analysis transform method to find the solution for modified Camassa–Holm and Degasperis–Procesi equations using the Caputo fractional operator.Both the considered equa...In the present study,we consider the q-homotopy analysis transform method to find the solution for modified Camassa–Holm and Degasperis–Procesi equations using the Caputo fractional operator.Both the considered equations are nonlinear and exemplify shallow water behaviour.We present the solution procedure for the fractional operator and the projected solution procedure gives a rapidly convergent series solution.The solution behaviour is demonstrated as compared with the exact solution and the response is plotted in 2D plots for a diverse fractional-order achieved by the Caputo derivative to show the importance of incorporating the generalised concept.The accuracy of the considered method is illustrated with available results in the numerical simulation.The convergence providence of the achieved solution is established in?-curves for a distinct arbitrary order.Moreover,some simulations and the important nature of the considered model,with the help of obtained results,shows the efficiency of the considered fractional operator and algorithm,while examining the nonlinear equations describing real-world problems.展开更多
The application of the q-homotopy analysis Shehu transform method(q-HAShTM)to discover the esti-mated solution of fractional Zakharov-Kuznetsov equations is investigated in this study.In the presence of a uniform magn...The application of the q-homotopy analysis Shehu transform method(q-HAShTM)to discover the esti-mated solution of fractional Zakharov-Kuznetsov equations is investigated in this study.In the presence of a uniform magnetic field,the Zakharov-Kuznetsov equations regulate the behaviour of nonlinear acoustic waves in a plasma containing cold ions and hot isothermal electrons.The q-HAShTM is a stable analytical method that combines homotopy analysis and the Shehu transform.This q-homotopy investigation Shehu transform is a constructive method that leads to the Zakharov-Kuznetsov equations,which regulate the propagation of nonlinear ion-acoustic waves in a plasma.It is a more semi-analytical method for adjust-ing and controlling the convergence region of the series solution and overcoming some of the homotopy analysis method’s limitations.展开更多
The study of internal atmospheric waves,also known as gravity waves,which are detectable inside the fluid rather than at the fluid surface,is presented in this work.We have used the time-fractional and fuzzy-fractiona...The study of internal atmospheric waves,also known as gravity waves,which are detectable inside the fluid rather than at the fluid surface,is presented in this work.We have used the time-fractional and fuzzy-fractional techniques to solve the differential equation system representing the atmospheric inter-nal waves model.The q-Homotopy analysis Shehu transform technique(q-HAShTM)is used to solve the model.The method helps find convergent solutions since it helps solve nonlinearity,and the fractional derivative can be easily computed using the Shehu transform.Finally,the obtained solution is compared for the particular case ofα=1 with the HAM solution to explain the method’s accuracy.展开更多
The study of dynamic behaviour of nonlinear models that arise in ocean engineering play a vital role in our daily life.There are many examples of ocean water waves which are nonlinear in nature.In shallow water,the li...The study of dynamic behaviour of nonlinear models that arise in ocean engineering play a vital role in our daily life.There are many examples of ocean water waves which are nonlinear in nature.In shallow water,the linearization of the equations imposes severe conditions on wave amplitude than it does in deep water,and the strong nonlinear effects are observed.In this paper,q-homotopy analysis Laplace transform scheme is used to inspect time dependent nonlinear Lane-Emden type equation of arbitrary order.It offers the solution in a fast converging series.The uniqueness and convergence analysis of the considered model is presented.The given examples confirm the competency as well as accuracy of the presented scheme.The behavior of obtained solution for distinct orders of fractional derivative is dis-cussed through graphs.The auxiliary parameter¯h offers a suitable mode of handling the region of con-vergence.The outcomes reveal that the q-HATM is attractive,reliable,efficient and very effective.展开更多
In this paper,an efficient hybrid numerical scheme which is based on a joint venture of the q-homotopy analysis method and Sumudu transform is applied to investigate the time-fractional modified Degasperis-Procesi(DP)...In this paper,an efficient hybrid numerical scheme which is based on a joint venture of the q-homotopy analysis method and Sumudu transform is applied to investigate the time-fractional modified Degasperis-Procesi(DP)equation.The present study considers the Caputo fractional derivative.The fractional order modified DP model is very important and plays a great role in study of ocean engineering and science.The proposed scheme provides a beautiful opportunity for proper selection of the auxiliary parameter h and the asymptotic parameterρ(≥1)to handle mainly the differential equations of nonlinear nature.The offered scheme produces the solution in the shape of a convergent series in a large admissible domain which is helpful to regulate the region of convergence of a series solution.The proposed work computes the approximate analytical solution of the fractional modified DP equation systematically and also presents graphically the variation of the obtained solution for diverse values of the fractional parameterβ.展开更多
基金supported by the National Natural Science Foundation of China(Grant No.12271433)。
文摘In this paper,two types of fractional nonlinear equations in Caputo sense,time-fractional Newell–Whitehead equation(FNWE)and time-fractional generalized Hirota–Satsuma coupled KdV system(HS-cKdVS),are investigated by means of the q-homotopy analysis method(q-HAM).The approximate solutions of the proposed equations are constructed in the form of a convergent series and are compared with the corresponding exact solutions.Due to the presence of the auxiliary parameter h in this method,just a few terms of the series solution are required in order to obtain better approximation.For the sake of visualization,the numerical results obtained in this paper are graphically displayed with the help of Maple.
基金Prince Sattam bin Abdulaziz University in Saudi Arabia supported this research under Project Number PSAU/2024/01/99519.
文摘This study intends to examine the analytical solutions to the resulting one-dimensional differential equation of acancer tumor model in the frame of time-fractional order with the Caputo-fractional operator employing a highlyefficient methodology called the q-homotopy analysis transform method.So,the preferred approach effectivelyfound the analytic series solution of the proposed model.The procured outcomes of the present frameworkdemonstrated that this method is authentic for obtaining solutions to a time-fractional-order cancer model.Theresults achieved graphically specify that the concerned paradigm is dependent on arbitrary order and parametersand also disclose the competence of the proposed algorithm.
文摘In this paper,we analyze the behaviour of solution for the system exemplifying model of tumour invasion and metastasis by the help of q-homotopy analysis transform method(q-HATM)with the fractional operator.The analyzed model consists of a system of three nonlinear differential equations elucidating the activation and the migratory response of the degradation of the matrix,tumour cells and production of degradative enzymes by the tumour cells.The considered method is graceful amalgamations of q-homotopy analysis technique with Laplace transform(LT),and Caputo–Fabrizio(CF)fractional operator is hired in the present study.By using the fixed point theory,existence and uniqueness are demonstrated.To validate and present the effectiveness of the considered algorithm,we analyzed the considered system in terms of fractional order with time and space.The error analysis of the considered scheme is illustrated.The variations with small change time with respect to achieved results are effectively captured in plots.The obtained results confirm that the considered method is very efficient and highly methodical to analyze the behaviors of the system of fractional order differential equations.
文摘In this paper, we consider Variational Iteration Method （VIM） and q-Homotopy Analysis Method （q-HAM） to sotve me partial differential equation resulted from Fingero Imbibition phenomena in double phase flow through porous media. We further compare the results obtained here with the solution obtained in [ 12] using Adomian Decomposition Method. Numerical results are obtained, using Mathematica 9, to show the effectiveness of these methods to our choice of problem especially for suitable values ofh and n.
文摘In this paper, an approximate solution for the one-dimensional hyperbolic telegraph equation by using the q-homotopy analysis method （q-HAM） is proposed.The results shows that the convergence of the q- homotopy analysis method is more accurate than the convergence of the homotopy analysis method （HAM）.
文摘This work focuses on designing and analyzing a double parametric fuzzy Homotopy analysis approach with Shehu transform for the non-linear fuzzy time-fractional generalized Fisher’s equation(FTFGFE).A triangular fuzzy number is used to describe the Caputo fractional derivative(CFD)of order(0,1)that appears in the modeling problem.A novel double parametric form-based Homotopy analysis approach with its convergence analysis is introduced to examine the fuzzy velocities profiles at different spatial positions with crisp and fuzzy conditions.Additional examples are offered to demonstrate the method’s efficacy and viability.The resulting results are compared to otherα=1 results to validate the obtained results and to test the efficiency of the proposed method.The errors approximations are provided to support the suggested computing efficiency of the analytical method.
文摘In this paper,we find the solutions for fractional potential Korteweg-de Vries(p-KdV)and Benjamin equations using q-homotopy analysis transform method(q-HATM).The considered method is the mixture of q-homotopy analysis method and Laplace transform,and the Caputo fractional operator is considered in the present investigation.The projected solution procedure manipulates and controls the obtained results in a large admissible domain.Further,it offers a simple algorithm to adjust the convergence province of the obtained solution.To validate the q-HATM is accurate and reliable,the numerical simulations have been conducted for both equations and the outcomes are revealed through the plots and tables.Comparison between the obtained solutions with the exact solutions exhibits that,the considered method is efficient and effective in solving nonlinear problems associated with science and technology.
文摘The pivotal aim of the present investigation is to find an approximate analytical solution for the system of three fractional differential equations describing the Lakes pollution using q-homotopy analysis transform method(q-HATM).We consider three different cases of the considered model namely,periodic input model,exponentially decaying input model,and linear input model.The considered scheme is unifications of q-homotopy analysis technique with Laplace transform(LT).To illustrate the existence and uniqueness for the projected model,we consider the fixed point hypothesis.More preciously,we scrutinized the behaviour of the obtained solution for the considered model with fractional-order,in order to elucidate the effectiveness of the proposed algorithm.Further,for the different fractional-order and parameters offered by the considered method,the physical natures have been apprehended.The obtained consequences evidence that the proposed method is very effective and highly methodical to study and examine the nature and its corresponding consequences of the system of fractional order differential equations describing the real word problems.
文摘The generalized Riccati equation mapping method(GREMM)is used in this paper to obtain different types of soliton solutions for nonlinear Schrödinger equation with higher dimension that existed in the regimes of anomalous dispersion.Later,we use the q-homotopy analysis method combined with the Laplace transform(q-HATM)to obtain approximate solutions of the bright and dark optical solitons.The q-HATM illustrates the solutions as a rapid convergent series.In addition,to show the physical behavior of the solutions obtained by the proposed techniques,the graphical representation has been provided with some parameter values.The findings demonstrate that the proposed techniques are useful,efficient and reliable mathematical method for the extraction of soliton solutions.
文摘In the present study,we consider the q-homotopy analysis transform method to find the solution for modified Camassa–Holm and Degasperis–Procesi equations using the Caputo fractional operator.Both the considered equations are nonlinear and exemplify shallow water behaviour.We present the solution procedure for the fractional operator and the projected solution procedure gives a rapidly convergent series solution.The solution behaviour is demonstrated as compared with the exact solution and the response is plotted in 2D plots for a diverse fractional-order achieved by the Caputo derivative to show the importance of incorporating the generalised concept.The accuracy of the considered method is illustrated with available results in the numerical simulation.The convergence providence of the achieved solution is established in?-curves for a distinct arbitrary order.Moreover,some simulations and the important nature of the considered model,with the help of obtained results,shows the efficiency of the considered fractional operator and algorithm,while examining the nonlinear equations describing real-world problems.
文摘The application of the q-homotopy analysis Shehu transform method(q-HAShTM)to discover the esti-mated solution of fractional Zakharov-Kuznetsov equations is investigated in this study.In the presence of a uniform magnetic field,the Zakharov-Kuznetsov equations regulate the behaviour of nonlinear acoustic waves in a plasma containing cold ions and hot isothermal electrons.The q-HAShTM is a stable analytical method that combines homotopy analysis and the Shehu transform.This q-homotopy investigation Shehu transform is a constructive method that leads to the Zakharov-Kuznetsov equations,which regulate the propagation of nonlinear ion-acoustic waves in a plasma.It is a more semi-analytical method for adjust-ing and controlling the convergence region of the series solution and overcoming some of the homotopy analysis method’s limitations.
文摘The study of internal atmospheric waves,also known as gravity waves,which are detectable inside the fluid rather than at the fluid surface,is presented in this work.We have used the time-fractional and fuzzy-fractional techniques to solve the differential equation system representing the atmospheric inter-nal waves model.The q-Homotopy analysis Shehu transform technique(q-HAShTM)is used to solve the model.The method helps find convergent solutions since it helps solve nonlinearity,and the fractional derivative can be easily computed using the Shehu transform.Finally,the obtained solution is compared for the particular case ofα=1 with the HAM solution to explain the method’s accuracy.
文摘The study of dynamic behaviour of nonlinear models that arise in ocean engineering play a vital role in our daily life.There are many examples of ocean water waves which are nonlinear in nature.In shallow water,the linearization of the equations imposes severe conditions on wave amplitude than it does in deep water,and the strong nonlinear effects are observed.In this paper,q-homotopy analysis Laplace transform scheme is used to inspect time dependent nonlinear Lane-Emden type equation of arbitrary order.It offers the solution in a fast converging series.The uniqueness and convergence analysis of the considered model is presented.The given examples confirm the competency as well as accuracy of the presented scheme.The behavior of obtained solution for distinct orders of fractional derivative is dis-cussed through graphs.The auxiliary parameter¯h offers a suitable mode of handling the region of con-vergence.The outcomes reveal that the q-HATM is attractive,reliable,efficient and very effective.
文摘In this paper,an efficient hybrid numerical scheme which is based on a joint venture of the q-homotopy analysis method and Sumudu transform is applied to investigate the time-fractional modified Degasperis-Procesi(DP)equation.The present study considers the Caputo fractional derivative.The fractional order modified DP model is very important and plays a great role in study of ocean engineering and science.The proposed scheme provides a beautiful opportunity for proper selection of the auxiliary parameter h and the asymptotic parameterρ(≥1)to handle mainly the differential equations of nonlinear nature.The offered scheme produces the solution in the shape of a convergent series in a large admissible domain which is helpful to regulate the region of convergence of a series solution.The proposed work computes the approximate analytical solution of the fractional modified DP equation systematically and also presents graphically the variation of the obtained solution for diverse values of the fractional parameterβ.
