Exact solutions of the (3+1)-dimensional space-time fractional Jimbo-Miwa equation are studied by... more Exact solutions of the (3+1)-dimensional space-time fractional Jimbo-Miwa equation are studied by the generalized Kudryashov method, the exp-function method and the (G′/G)-expansion method. The solutions obtained include the form of hyperbolic functions, trigonometric and rational functions. These methods are effective, simple, and many types of solutions can be obtained at the same time.
In this paper, we pay attention to the analytical method named, ansatz method for finding the exa... more In this paper, we pay attention to the analytical method named, ansatz method for finding the exact solutions of the variable-coefficient modified KdV equation and variable coefficient diffusion-reaction equation. As a result the singular 1-soliton solution is obtained. These solutions are important for the explanation of some practical physical problems. The obtained results show that these methods provides a powerful mathematical tool for solving nonlinear equations with variable coefficients. This method can be extended to solve other variable coefficient nonlinear partial differential equations.
In this paper, we obtain exact solutions of two time fractional differential equations using Juma... more In this paper, we obtain exact solutions of two time fractional differential equations using Jumarie’s modified Riemann-Liouville derivative which is encountered in mathematical physics and applied mathematics; namely (3 + 1)-dimensional time fractional KdV-ZK equation and time fractional ADR equation by using fractional complex transform and (G′G)-expansion method. It is shown that the considered transform and method are very useful in solving nonlinear fractional differential equations.
In this paper, we will implement a comparison between three different perceptions to the behavior... more In this paper, we will implement a comparison between three different perceptions to the behavior of the exact solution to the generalized Hirota–Satsuma coupling Korteweg–de Vries System (GHSCKDVS). The proposed model plays a vital role in different branches of physics and applied mathematics. Specially, this model describes dispersive waves as the shallow water waves in hydrodynamics and it has various types of solutions like soliton solutions and traveling wave solutions. These three various visions to the behavior of the exact solution are established via three various techniques that have three different mathematical analyses. These three important different techniques are the Paul–Painlevé approach method (PPAM) examined previously to extract the exact solutions for many nonlinear partial differential equations (NLPDE) and achieved good results, the Ricatti–Bernolli Sub-ODE method (RBSODM) which is one of the famous ansatze approaches that doesn’t surrender to the balance rule, treats the balance rule fails and realizes impressive forms of the exact solutions. Many new various types of soliton solutions as hyperbolic and trigonometric function solutions, represented by the M-shaped, W-shaped, kink, anti-kink and dark soliton solutions have been established via these two methods. Moreover, we will document the identical numerical solutions for all achieved exact solutions realized by the two methods mentioned above via the famous numerical variational iteration method (VIM) that usually gives good results. The powerful of our achieved numerical solutions related to that their initial conditions emerged from the extracted exact solutions.
Journal of the Association of Arab Universities for Basic and Applied Sciences, Oct 1, 2017
Using the Exp-function method, we derive exact solutions of the nonlinear space-time fractional T... more Using the Exp-function method, we derive exact solutions of the nonlinear space-time fractional Telegraph equation and space-time fractional KPP equation. As a result, we obtain many exact analytical solutions including hyperbolic function. The fractional derivative is described in Jumarie's modified Riemann-Liouville sense. This method is very effective and convenient for solving nonlinear fractional differential equations.
Advances in Applied Mathematics and Mechanics, Jan 27, 2016
In this paper, the (G ′ /G)-expansion method is suggested to establish new exact solutions for fr... more In this paper, the (G ′ /G)-expansion method is suggested to establish new exact solutions for fractional differential-difference equations in the sense of modified Riemann-Liouville derivative. The fractional complex transform is proposed to convert a fractional partial differential difference equation into its differential difference equation of integer order. With the aid of symbolic computation, we choose nonlinear lattice equations to illustrate the validity and advantages of the algorithm. It is shown that the proposed algorithm is effective and can be used for many other nonlinear lattice equations in mathematical physics and applied mathematics.
In this paper, we study the exact solutions of nonlinear time fractional Clannish Random Walker's... more In this paper, we study the exact solutions of nonlinear time fractional Clannish Random Walker's Parabolic (CRWP) equation. We extend the G 0 G and (G 0 =G; 1=G)-expansion methods to fractional di¤erential equations in the sense of modi…ed Riemann-Liouville derivative based on fractional complex transformation. We obtained hyperbolic function solutions, trigonometric function solutions and rational function solutions. It was shown that the considered methods and transform are very reliable and e¢ cient for these type fractional equations. These methods and transform can be used in studying many other nonlinear time and space fractional di¤erential equations and nonlinear systems.
Communications in Theoretical Physics, Feb 1, 2017
Dark soliton solutions for space-time fractional Sharma–Tasso–Olver and space-time fractional pot... more Dark soliton solutions for space-time fractional Sharma–Tasso–Olver and space-time fractional potential Kadomtsev–Petviashvili equations are determined by using the properties of modified Riemann–Liouville derivative and fractional complex transform. After reducing both equations to nonlinear ODEs with constant coefficients, the ansatz is substituted into the resultant nonlinear ODEs. The coefficients of the solutions in the ansatz are calculated by algebraic computer computations. Two different solutions are obtained for the Sharma–Tasso–Olver equation as only one solution for the potential Kadomtsev–Petviashvili equation. The solution profiles are demonstrated in 3D plots in finite domains of time and space.
In this work, we propose a new approach, namely ansatz method, for solving fractional differentia... more In this work, we propose a new approach, namely ansatz method, for solving fractional differential equations based on a fractional complex transform and apply it to the nonlinear partial space–time fractional modified Benjamin–Bona–Mahoney(m BBM) equation, the time fractional m Kd V equation and the nonlinear fractional Zoomeron equation which gives rise to some new exact solutions. The physical parameters in the soliton solutions: amplitude, inverse width, free parameters and velocity are obtained as functions of the dependent model coefficients. This method is suitable and more powerful for solving other kinds of nonlinear fractional PDEs arising in mathematical physics. Since the fractional derivatives are described in the modified Riemann–Liouville sense.
In this paper, the fractional complex transform and the (G ′ G) expansion method are employed to ... more In this paper, the fractional complex transform and the (G ′ G) expansion method are employed to solve the time-fractional modfied Kortewegde Vries equation (fmKdV), Sharma-Tasso-Olver, Fitzhugh-Nagumo equations, where G satisfies a second order linear ordinary differential equation. Exact solutions are expressed in terms of hyperbolic, trigonometric and rational functions. These solutions may be useful and desirable to explain some nonlinear physical phenomena in genuinely nonlinear fractional calculus.
In this work, we consider the (1+1)-dimensional modified improved Boussinesq (IMBq) equation. As ... more In this work, we consider the (1+1)-dimensional modified improved Boussinesq (IMBq) equation. As the considered equation is of evolution type, no recourse to a Lagrangian formulation is made. However, we showed that by utilising the partial Lagrangian method and multiplier method, one can construct a number of local and nonlocal conservation laws for the IMBq equation. In addition, by using a solitary wave ansatz method, we obtained exact bright soliton solutions for this equation. The parameters of the soliton envelope (amplitude, widths, velocity) were obtained as function of the dependent model coefficients. Note that, it is always useful and desirable to construct exact solutions especially soliton-type envelope for the understanding of most nonlinear physical phenomena.
This article utilizes the local fractional derivative and the expfunction method to construct the... more This article utilizes the local fractional derivative and the expfunction method to construct the exact solutions of nonlinear time-fractional differential equations (FDEs). For illustrating the validity of the method, it is applied to the time-fractional Camassa-Holm equation and the time-fractional-generalized fifth-order KdV equation. Moreover, the exact solutions are obtained for the equations which are formed by different parameter values related to the time-fractional-generalized fifth-order KdV equation. This method is an reliable and efficient mathematical tool for solving FDEs and it can be applied to other non-linear FDEs.
In this article, we establish exact solutions for variable-coefficient modified KdV equation, var... more In this article, we establish exact solutions for variable-coefficient modified KdV equation, variable-coefficient KdV equation, and variable-coefficient diffusion–reaction equations. The modified sine-cosine method is used to construct exact periodic solutions. These solutions may be important for the explanation of some practical physical problems. The obtained results show that the modified sine-cosine method provides a powerful mathematical tool for solving nonlinear equations with variable coefficients.
In this work, we …nd exact solutions of space-time fractional Klein-Gordon equation including Jum... more In this work, we …nd exact solutions of space-time fractional Klein-Gordon equation including Jumarie's modi…ed Riemann-Liouville derivative by using fractional complex transform, (G 0 =G) and (w=g)-expansion methods which is employed to solve fractional di¤erential equations in mathematical physics and applied mathematics. As a result, two types of exact solutions are obtained. It is shown that the considered transform and method are very e¢ cient and powerful in solving other kinds of nonlinear fractional di¤erential equations.
The Fitzhugh–Nagumo equation is an important nonlinear reaction-diffusion equation used to model ... more The Fitzhugh–Nagumo equation is an important nonlinear reaction-diffusion equation used to model the transmission of nerve impulses. This equation is used in biology as population genetics, the Fitzhugh–Nagumo equation is also frequently used in circuit theory. In this study, we gave solutions to the fractional Fitzhugh–Nagumo (FN) equation, the fractional Newell–Whitehead–Segel (NWS) equation, and the fractional Zeldovich equation. We have obtained exact solutions within time fractional conformable derivative for these equations.
ABSTRACT This paper presents an approach for solving fractional differential equations by employi... more ABSTRACT This paper presents an approach for solving fractional differential equations by employing the exp-function method and (G'/G)-expansion method. These methods were applied in two examples to solve non-linear fractional differential equations. The fractional derivatives are described in the modified Riemann-Liouville sense. As a result, many exact analytical solutions are obtained including hyperbolic function solutions and trigonometric function solutions. The results also show that the methods are very effective and convenient for solving nonlinear partial differential equations of fractional order.
DOAJ (DOAJ: Directory of Open Access Journals), Dec 1, 2015
In this paper, we obtained the 1-soliton solutions of the (3+1)-dimensional Zakharov-Kuznetsov (Z... more In this paper, we obtained the 1-soliton solutions of the (3+1)-dimensional Zakharov-Kuznetsov (ZK) equation and the Drinfeld-Sokolov equation system. By using a solitary wave ansatz method, we obtain exact bright and dark soliton solutions for these equations. The parameters of the soliton envelope (amplitude, widths, velocity) are obtained as function of the dependent model coefficients. Note that, it is always useful and desirable to construct exact solutions especially soliton-type envelope for the understanding of most nonlinear physical phenomena.
A shock wave is a propagating disturbance that travels through a medium. It actually represents a... more A shock wave is a propagating disturbance that travels through a medium. It actually represents a sharp discontinuity of the parameters that describe the medium, namely pressure, temperature and density. In this paper, shock wave solutions are obtained for the (2+1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation using a solitary wave ansatz. Constraint relations for this equation are also elucidated for the existence of shock wave.
In this paper, we obtained the 1-soliton solutions of the (3+1)-dimensional Zakharov-Kuznetsov (Z... more In this paper, we obtained the 1-soliton solutions of the (3+1)-dimensional Zakharov-Kuznetsov (ZK) equation and the Drinfeld-Sokolov equation system. By using a solitary wave ansatz method, we obtain exact bright and dark soliton solutions for these equations. The parameters of the soliton envelope (amplitude, widths, velocity) are obtained as function of the dependent model coefficients. Note that, it is always useful and desirable to construct exact solutions especially soliton-type envelope for the understanding of most nonlinear physical phenomena.
In this article, we use the modified Riemann Liouville derivative and fractional complex transfor... more In this article, we use the modified Riemann Liouville derivative and fractional complex transform for converting the time fractional equation into its corresponding ordinary differential equation. We dealt with exact solutions of the seventh-order time fractional Sawada-Kotera-Ito equation. We obtained exact solutions of the given equation via the exp-function method, the (G /G)-expansion method and the ansatz method. By using these methods, dark, bright and singular soliton solutions of this equation have been found. The physical parameters in the dark, bright and singular soliton solutions free parameters and velocity are obtained as functions of the dependent model coefficients. Finally, some graphical representations of this equation are derived. It is shown that these methods are further efficient, convenient, and can be used to establish new solutions for other kind of nonlinear fractional differential equations arising in mathematical physics.
Exact solutions of the (3+1)-dimensional space-time fractional Jimbo-Miwa equation are studied by... more Exact solutions of the (3+1)-dimensional space-time fractional Jimbo-Miwa equation are studied by the generalized Kudryashov method, the exp-function method and the (G′/G)-expansion method. The solutions obtained include the form of hyperbolic functions, trigonometric and rational functions. These methods are effective, simple, and many types of solutions can be obtained at the same time.
In this paper, we pay attention to the analytical method named, ansatz method for finding the exa... more In this paper, we pay attention to the analytical method named, ansatz method for finding the exact solutions of the variable-coefficient modified KdV equation and variable coefficient diffusion-reaction equation. As a result the singular 1-soliton solution is obtained. These solutions are important for the explanation of some practical physical problems. The obtained results show that these methods provides a powerful mathematical tool for solving nonlinear equations with variable coefficients. This method can be extended to solve other variable coefficient nonlinear partial differential equations.
In this paper, we obtain exact solutions of two time fractional differential equations using Juma... more In this paper, we obtain exact solutions of two time fractional differential equations using Jumarie’s modified Riemann-Liouville derivative which is encountered in mathematical physics and applied mathematics; namely (3 + 1)-dimensional time fractional KdV-ZK equation and time fractional ADR equation by using fractional complex transform and (G′G)-expansion method. It is shown that the considered transform and method are very useful in solving nonlinear fractional differential equations.
In this paper, we will implement a comparison between three different perceptions to the behavior... more In this paper, we will implement a comparison between three different perceptions to the behavior of the exact solution to the generalized Hirota–Satsuma coupling Korteweg–de Vries System (GHSCKDVS). The proposed model plays a vital role in different branches of physics and applied mathematics. Specially, this model describes dispersive waves as the shallow water waves in hydrodynamics and it has various types of solutions like soliton solutions and traveling wave solutions. These three various visions to the behavior of the exact solution are established via three various techniques that have three different mathematical analyses. These three important different techniques are the Paul–Painlevé approach method (PPAM) examined previously to extract the exact solutions for many nonlinear partial differential equations (NLPDE) and achieved good results, the Ricatti–Bernolli Sub-ODE method (RBSODM) which is one of the famous ansatze approaches that doesn’t surrender to the balance rule, treats the balance rule fails and realizes impressive forms of the exact solutions. Many new various types of soliton solutions as hyperbolic and trigonometric function solutions, represented by the M-shaped, W-shaped, kink, anti-kink and dark soliton solutions have been established via these two methods. Moreover, we will document the identical numerical solutions for all achieved exact solutions realized by the two methods mentioned above via the famous numerical variational iteration method (VIM) that usually gives good results. The powerful of our achieved numerical solutions related to that their initial conditions emerged from the extracted exact solutions.
Journal of the Association of Arab Universities for Basic and Applied Sciences, Oct 1, 2017
Using the Exp-function method, we derive exact solutions of the nonlinear space-time fractional T... more Using the Exp-function method, we derive exact solutions of the nonlinear space-time fractional Telegraph equation and space-time fractional KPP equation. As a result, we obtain many exact analytical solutions including hyperbolic function. The fractional derivative is described in Jumarie's modified Riemann-Liouville sense. This method is very effective and convenient for solving nonlinear fractional differential equations.
Advances in Applied Mathematics and Mechanics, Jan 27, 2016
In this paper, the (G ′ /G)-expansion method is suggested to establish new exact solutions for fr... more In this paper, the (G ′ /G)-expansion method is suggested to establish new exact solutions for fractional differential-difference equations in the sense of modified Riemann-Liouville derivative. The fractional complex transform is proposed to convert a fractional partial differential difference equation into its differential difference equation of integer order. With the aid of symbolic computation, we choose nonlinear lattice equations to illustrate the validity and advantages of the algorithm. It is shown that the proposed algorithm is effective and can be used for many other nonlinear lattice equations in mathematical physics and applied mathematics.
In this paper, we study the exact solutions of nonlinear time fractional Clannish Random Walker's... more In this paper, we study the exact solutions of nonlinear time fractional Clannish Random Walker's Parabolic (CRWP) equation. We extend the G 0 G and (G 0 =G; 1=G)-expansion methods to fractional di¤erential equations in the sense of modi…ed Riemann-Liouville derivative based on fractional complex transformation. We obtained hyperbolic function solutions, trigonometric function solutions and rational function solutions. It was shown that the considered methods and transform are very reliable and e¢ cient for these type fractional equations. These methods and transform can be used in studying many other nonlinear time and space fractional di¤erential equations and nonlinear systems.
Communications in Theoretical Physics, Feb 1, 2017
Dark soliton solutions for space-time fractional Sharma–Tasso–Olver and space-time fractional pot... more Dark soliton solutions for space-time fractional Sharma–Tasso–Olver and space-time fractional potential Kadomtsev–Petviashvili equations are determined by using the properties of modified Riemann–Liouville derivative and fractional complex transform. After reducing both equations to nonlinear ODEs with constant coefficients, the ansatz is substituted into the resultant nonlinear ODEs. The coefficients of the solutions in the ansatz are calculated by algebraic computer computations. Two different solutions are obtained for the Sharma–Tasso–Olver equation as only one solution for the potential Kadomtsev–Petviashvili equation. The solution profiles are demonstrated in 3D plots in finite domains of time and space.
In this work, we propose a new approach, namely ansatz method, for solving fractional differentia... more In this work, we propose a new approach, namely ansatz method, for solving fractional differential equations based on a fractional complex transform and apply it to the nonlinear partial space–time fractional modified Benjamin–Bona–Mahoney(m BBM) equation, the time fractional m Kd V equation and the nonlinear fractional Zoomeron equation which gives rise to some new exact solutions. The physical parameters in the soliton solutions: amplitude, inverse width, free parameters and velocity are obtained as functions of the dependent model coefficients. This method is suitable and more powerful for solving other kinds of nonlinear fractional PDEs arising in mathematical physics. Since the fractional derivatives are described in the modified Riemann–Liouville sense.
In this paper, the fractional complex transform and the (G ′ G) expansion method are employed to ... more In this paper, the fractional complex transform and the (G ′ G) expansion method are employed to solve the time-fractional modfied Kortewegde Vries equation (fmKdV), Sharma-Tasso-Olver, Fitzhugh-Nagumo equations, where G satisfies a second order linear ordinary differential equation. Exact solutions are expressed in terms of hyperbolic, trigonometric and rational functions. These solutions may be useful and desirable to explain some nonlinear physical phenomena in genuinely nonlinear fractional calculus.
In this work, we consider the (1+1)-dimensional modified improved Boussinesq (IMBq) equation. As ... more In this work, we consider the (1+1)-dimensional modified improved Boussinesq (IMBq) equation. As the considered equation is of evolution type, no recourse to a Lagrangian formulation is made. However, we showed that by utilising the partial Lagrangian method and multiplier method, one can construct a number of local and nonlocal conservation laws for the IMBq equation. In addition, by using a solitary wave ansatz method, we obtained exact bright soliton solutions for this equation. The parameters of the soliton envelope (amplitude, widths, velocity) were obtained as function of the dependent model coefficients. Note that, it is always useful and desirable to construct exact solutions especially soliton-type envelope for the understanding of most nonlinear physical phenomena.
This article utilizes the local fractional derivative and the expfunction method to construct the... more This article utilizes the local fractional derivative and the expfunction method to construct the exact solutions of nonlinear time-fractional differential equations (FDEs). For illustrating the validity of the method, it is applied to the time-fractional Camassa-Holm equation and the time-fractional-generalized fifth-order KdV equation. Moreover, the exact solutions are obtained for the equations which are formed by different parameter values related to the time-fractional-generalized fifth-order KdV equation. This method is an reliable and efficient mathematical tool for solving FDEs and it can be applied to other non-linear FDEs.
In this article, we establish exact solutions for variable-coefficient modified KdV equation, var... more In this article, we establish exact solutions for variable-coefficient modified KdV equation, variable-coefficient KdV equation, and variable-coefficient diffusion–reaction equations. The modified sine-cosine method is used to construct exact periodic solutions. These solutions may be important for the explanation of some practical physical problems. The obtained results show that the modified sine-cosine method provides a powerful mathematical tool for solving nonlinear equations with variable coefficients.
In this work, we …nd exact solutions of space-time fractional Klein-Gordon equation including Jum... more In this work, we …nd exact solutions of space-time fractional Klein-Gordon equation including Jumarie's modi…ed Riemann-Liouville derivative by using fractional complex transform, (G 0 =G) and (w=g)-expansion methods which is employed to solve fractional di¤erential equations in mathematical physics and applied mathematics. As a result, two types of exact solutions are obtained. It is shown that the considered transform and method are very e¢ cient and powerful in solving other kinds of nonlinear fractional di¤erential equations.
The Fitzhugh–Nagumo equation is an important nonlinear reaction-diffusion equation used to model ... more The Fitzhugh–Nagumo equation is an important nonlinear reaction-diffusion equation used to model the transmission of nerve impulses. This equation is used in biology as population genetics, the Fitzhugh–Nagumo equation is also frequently used in circuit theory. In this study, we gave solutions to the fractional Fitzhugh–Nagumo (FN) equation, the fractional Newell–Whitehead–Segel (NWS) equation, and the fractional Zeldovich equation. We have obtained exact solutions within time fractional conformable derivative for these equations.
ABSTRACT This paper presents an approach for solving fractional differential equations by employi... more ABSTRACT This paper presents an approach for solving fractional differential equations by employing the exp-function method and (G'/G)-expansion method. These methods were applied in two examples to solve non-linear fractional differential equations. The fractional derivatives are described in the modified Riemann-Liouville sense. As a result, many exact analytical solutions are obtained including hyperbolic function solutions and trigonometric function solutions. The results also show that the methods are very effective and convenient for solving nonlinear partial differential equations of fractional order.
DOAJ (DOAJ: Directory of Open Access Journals), Dec 1, 2015
In this paper, we obtained the 1-soliton solutions of the (3+1)-dimensional Zakharov-Kuznetsov (Z... more In this paper, we obtained the 1-soliton solutions of the (3+1)-dimensional Zakharov-Kuznetsov (ZK) equation and the Drinfeld-Sokolov equation system. By using a solitary wave ansatz method, we obtain exact bright and dark soliton solutions for these equations. The parameters of the soliton envelope (amplitude, widths, velocity) are obtained as function of the dependent model coefficients. Note that, it is always useful and desirable to construct exact solutions especially soliton-type envelope for the understanding of most nonlinear physical phenomena.
A shock wave is a propagating disturbance that travels through a medium. It actually represents a... more A shock wave is a propagating disturbance that travels through a medium. It actually represents a sharp discontinuity of the parameters that describe the medium, namely pressure, temperature and density. In this paper, shock wave solutions are obtained for the (2+1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation using a solitary wave ansatz. Constraint relations for this equation are also elucidated for the existence of shock wave.
In this paper, we obtained the 1-soliton solutions of the (3+1)-dimensional Zakharov-Kuznetsov (Z... more In this paper, we obtained the 1-soliton solutions of the (3+1)-dimensional Zakharov-Kuznetsov (ZK) equation and the Drinfeld-Sokolov equation system. By using a solitary wave ansatz method, we obtain exact bright and dark soliton solutions for these equations. The parameters of the soliton envelope (amplitude, widths, velocity) are obtained as function of the dependent model coefficients. Note that, it is always useful and desirable to construct exact solutions especially soliton-type envelope for the understanding of most nonlinear physical phenomena.
In this article, we use the modified Riemann Liouville derivative and fractional complex transfor... more In this article, we use the modified Riemann Liouville derivative and fractional complex transform for converting the time fractional equation into its corresponding ordinary differential equation. We dealt with exact solutions of the seventh-order time fractional Sawada-Kotera-Ito equation. We obtained exact solutions of the given equation via the exp-function method, the (G /G)-expansion method and the ansatz method. By using these methods, dark, bright and singular soliton solutions of this equation have been found. The physical parameters in the dark, bright and singular soliton solutions free parameters and velocity are obtained as functions of the dependent model coefficients. Finally, some graphical representations of this equation are derived. It is shown that these methods are further efficient, convenient, and can be used to establish new solutions for other kind of nonlinear fractional differential equations arising in mathematical physics.
Uploads
Papers by Özkan Güner