Epidemiological models have been playing a vital role in different areas of biological sciences f... more Epidemiological models have been playing a vital role in different areas of biological sciences for the analysis of various contagious diseases. Transmissibility of virulent diseases is being portrayed in the literature through different compartments such as susceptible, infected, recovered (SIR), susceptible, infected, recovered, susceptible (SIRS) or susceptible, exposed, infected, recovered (SEIR), etc. The novelty in this endeavor is the addition of compartments of latency and treatment with vaccination, so the system is designated as susceptible, vaccinated, exposed, latent, infected, treatment, and recovered (SVELITR). The contact of a susceptible individual to an infective individual firstly makes the individual exposed, latent, and then completely infection carrier. Innovatively, the assumption that exposed, latent, and infected individuals enter the treatment compartment at different rates after a time lag is also deliberated through the existence of time delay. The rate of...
The motivation of this paper is to study the fourth order Emden-Fowler equations with initial val... more The motivation of this paper is to study the fourth order Emden-Fowler equations with initial values by Haarwavelet collocation method(HWCM). In this methodology, differential equations are transformed into a system of linear or nonlinear equations that leads to the value of Haar coefficients and later the solution can be obtained on the entire domain (0, 1]. The fourth order nonlinear test examples are solved at different Haar levels to analyze the accuracy of results. The simplicity and effectiveness of the method made it more attractive than non-perturbative methods found in literature.
In this paper, fuzzy fractional diffusion equations (FFDEs) and fuzzy fractional wave equations (... more In this paper, fuzzy fractional diffusion equations (FFDEs) and fuzzy fractional wave equations (FFWEs), subjected to initial and boundary conditions are considered. As these equations have significant applications in physics and engineering, a methodical spectral-tau scheme is utilized to obtain efficient solutions of FFDE and FFWE. For this purpose, shifted Chebyshev polynomials (SCPs) together with its operational matrix of integration in Riemann-Liouville sense and operation matrix of derivative in Caputo sense are employed to approximate the fuzzy-valued functions, their integral and differential terms, respectively. The proposed method is applied to some illustrative examples considered under generalized Hukuhara partial differentiability (ɡHP-differentiability). Graphical results are included with error bar plots of each example that show the efficiency and convergence of the method towards the exact solution.
The main objective of the present investigation is to examine the couple stress fluid flow occurr... more The main objective of the present investigation is to examine the couple stress fluid flow occurring as a result of rotation of a disk. On implementing a suitable transformation, the governing system of partial differential equations (PDEs) is converted into nonlinear differential equations of a single independent variable. These equations are solved analytically by virtue of the Homotopy Analysis Method (HAM) which gives solutions in the form of a series. The solution of most of the governing problems is determined in terms of the absolute exponential and decaying functions by means of this powerful technique. To support analytic results, some graphs are plotted for determining the convergence of the solution. Also the graphical interpretation of velocity profiles corresponding to the effects of pertinent parameters are shown and discussed in detail. The numerical results are calculated for evaluation of the influence of fluid parameter. It can also be anticipated that the radial a...
This paper provides analytical solution of the non-aligned stagnation point flow of second grade ... more This paper provides analytical solution of the non-aligned stagnation point flow of second grade fluid over a porous rotating disk in the presence of a magnetic field and suction/injection at the disk surface. The mathematical formulation of the fluid model is obtained in terms of partial differential equations (PDEs). The PDEs governing the motion are transformed into a system of ordinary differential equations (ODEs) by means of a similarity transformation and these corresponding nonlinear ODEs are solved by employing the homotopy analysis method (HAM) and the convergence analysis of the presented method is also performed graphically. An inclusion of the influences of various admissible parameters has been shown numerically and graphically on the flow field. Furthermore, comparison is made and it concedes that the obtained results are found to be in good agreement with results existing in literature.
This article presents the study of three-dimensional, stagnation flow of a Powell-Eyring fluid to... more This article presents the study of three-dimensional, stagnation flow of a Powell-Eyring fluid towards an off-centered rotating porous disk. A uniform injection or suction is applied through the surface of the disk. The Darcy law of porous disk for Powell-Eyring fluid is also obtained. The governing partial differential equations and their related boundary conditions are converted into ordinary differential equations by using a suitable similarity transformation. The analytical solution, of the system of equations is solved by using homotopy analysis method. The convergence region of the obtained solution is determined and plotted. The effects of rotational parameter, porosity of the medium, the characteristics of the non-Newtonian fluid and the suction or injection velocity on the velocity distributions is shown by graphical representation.
Annals of West University of Timisoara - Mathematics and Computer Science, 2018
Bearing in mind the considerable importance of fuzzy differential equations (FDEs) in different f... more Bearing in mind the considerable importance of fuzzy differential equations (FDEs) in different fields of science and engineering, in this paper, nonlinear nth order FDEs are approximated, heuristically. The analysis is carried out on using Chebyshev neural network (ChNN), which is a type of single layer functional link artificial neural network (FLANN). Besides, explication of generalized Hukuhara differentiability (gH-differentiability) is also added for the nth order differentiability of fuzzy-valued functions. Moreover, general formulation of the structure of ChNN for the governing problem is described and assessed on some examples of nonlinear FDEs. In addition, comparison analysis of the proposed method with Runge-Kutta method is added and also portrayed the error bars that clarify the feasibility of attained solutions and validity of the method.
In this paper, an extension is introduced into Max-Min Improved Euler methods for solving initial... more In this paper, an extension is introduced into Max-Min Improved Euler methods for solving initial value problems of fuzzy fractional differential equations (FFDEs). Two modified fractional Euler type methods have been proposed and investigated to obtain numerical solutions of linear and nonlinear FFDEs. The proposed algorithms are tested on various illustrative examples. Exact values are also simulated to compare and discuss the closeness and accuracy of approximations so obtained. Comparatively, tables and graphs results reveal the complete reliability, efficiency and accuracy of the proposed methods.
In this paper the helical flows of fractionalized Maxwell fluid model, through a circular cylinde... more In this paper the helical flows of fractionalized Maxwell fluid model, through a circular cylinder, is studied. The motion is produced by the cylinder that at the initial moment begins to rotate around its axis with an angular velocity Omegatp, and to slide along the same axis with linear velocity Utp. The solutions that have been obtained using Laplace and finite Hankel transforms and presented in series form in terms of the newly defined special function M(z), satisfy all imposed initial and boundary conditions. Moreover, the corresponding solutions for ordinary Maxwell and Newtonian fluid obtained as special cases of the present general solution. Finally, the influence of various pertinent parameters on fluid motion as well as the comparison among different fluids models is analyzed by graphical illustrations.
International Journal of Differential Equations, 2011
We applied a new approach to obtain natural frequency of the nonlinear oscillator with discontinu... more We applied a new approach to obtain natural frequency of the nonlinear oscillator with discontinuity. He's Hamiltonian approach is modified for nonlinear oscillator with discontinuity for which the elastic force term is proportional to sgn(u). We employed this method for higher-order approximate solution of the nonlinear oscillator equation. This property is used to obtain approximate frequency-amplitude relationship of a nonlinear oscillator with high accuracy. Many numerical results are given to prove the efficiency of the suggested technique.
We applied an approach to obtain the natural frequency of the generalized Duffing oscillatoru¨+u+... more We applied an approach to obtain the natural frequency of the generalized Duffing oscillatoru¨+u+α3u3+α5u5+α7u7+⋯+αnun=0and a nonlinear oscillator with a restoring force which is the function of a noninteger power exponent of deflectionu¨+αu|u|n−1=0. This approach is based on involved parameters, initial conditions, and collocation points. For any arbitrary power ofn, the approximate frequency analysis is carried out between the natural frequency and amplitude. The solution procedure is simple, and the results obtained are valid for the whole solution domain.
Abstract In this article, we study numerical solutions of time-fractional fourth-order partial di... more Abstract In this article, we study numerical solutions of time-fractional fourth-order partial differential equations with variable coefficients by introducing the fractional derivative in the sense of Caputo. We implement reliable series solution techniques namely Adomian decomposition method (ADM) and He’s variational iteration method (HVIM). Some applications are presented to highlight the significant features of these techniques. The comparison shows that the solutions obtained are in good agreement with each other and with their respective exact solutions. Some of these types of differential equations arise practically in the theory of transverse vibrations.
Epidemiological models have been playing a vital role in different areas of biological sciences f... more Epidemiological models have been playing a vital role in different areas of biological sciences for the analysis of various contagious diseases. Transmissibility of virulent diseases is being portrayed in the literature through different compartments such as susceptible, infected, recovered (SIR), susceptible, infected, recovered, susceptible (SIRS) or susceptible, exposed, infected, recovered (SEIR), etc. The novelty in this endeavor is the addition of compartments of latency and treatment with vaccination, so the system is designated as susceptible, vaccinated, exposed, latent, infected, treatment, and recovered (SVELITR). The contact of a susceptible individual to an infective individual firstly makes the individual exposed, latent, and then completely infection carrier. Innovatively, the assumption that exposed, latent, and infected individuals enter the treatment compartment at different rates after a time lag is also deliberated through the existence of time delay. The rate of...
The motivation of this paper is to study the fourth order Emden-Fowler equations with initial val... more The motivation of this paper is to study the fourth order Emden-Fowler equations with initial values by Haarwavelet collocation method(HWCM). In this methodology, differential equations are transformed into a system of linear or nonlinear equations that leads to the value of Haar coefficients and later the solution can be obtained on the entire domain (0, 1]. The fourth order nonlinear test examples are solved at different Haar levels to analyze the accuracy of results. The simplicity and effectiveness of the method made it more attractive than non-perturbative methods found in literature.
In this paper, fuzzy fractional diffusion equations (FFDEs) and fuzzy fractional wave equations (... more In this paper, fuzzy fractional diffusion equations (FFDEs) and fuzzy fractional wave equations (FFWEs), subjected to initial and boundary conditions are considered. As these equations have significant applications in physics and engineering, a methodical spectral-tau scheme is utilized to obtain efficient solutions of FFDE and FFWE. For this purpose, shifted Chebyshev polynomials (SCPs) together with its operational matrix of integration in Riemann-Liouville sense and operation matrix of derivative in Caputo sense are employed to approximate the fuzzy-valued functions, their integral and differential terms, respectively. The proposed method is applied to some illustrative examples considered under generalized Hukuhara partial differentiability (ɡHP-differentiability). Graphical results are included with error bar plots of each example that show the efficiency and convergence of the method towards the exact solution.
The main objective of the present investigation is to examine the couple stress fluid flow occurr... more The main objective of the present investigation is to examine the couple stress fluid flow occurring as a result of rotation of a disk. On implementing a suitable transformation, the governing system of partial differential equations (PDEs) is converted into nonlinear differential equations of a single independent variable. These equations are solved analytically by virtue of the Homotopy Analysis Method (HAM) which gives solutions in the form of a series. The solution of most of the governing problems is determined in terms of the absolute exponential and decaying functions by means of this powerful technique. To support analytic results, some graphs are plotted for determining the convergence of the solution. Also the graphical interpretation of velocity profiles corresponding to the effects of pertinent parameters are shown and discussed in detail. The numerical results are calculated for evaluation of the influence of fluid parameter. It can also be anticipated that the radial a...
This paper provides analytical solution of the non-aligned stagnation point flow of second grade ... more This paper provides analytical solution of the non-aligned stagnation point flow of second grade fluid over a porous rotating disk in the presence of a magnetic field and suction/injection at the disk surface. The mathematical formulation of the fluid model is obtained in terms of partial differential equations (PDEs). The PDEs governing the motion are transformed into a system of ordinary differential equations (ODEs) by means of a similarity transformation and these corresponding nonlinear ODEs are solved by employing the homotopy analysis method (HAM) and the convergence analysis of the presented method is also performed graphically. An inclusion of the influences of various admissible parameters has been shown numerically and graphically on the flow field. Furthermore, comparison is made and it concedes that the obtained results are found to be in good agreement with results existing in literature.
This article presents the study of three-dimensional, stagnation flow of a Powell-Eyring fluid to... more This article presents the study of three-dimensional, stagnation flow of a Powell-Eyring fluid towards an off-centered rotating porous disk. A uniform injection or suction is applied through the surface of the disk. The Darcy law of porous disk for Powell-Eyring fluid is also obtained. The governing partial differential equations and their related boundary conditions are converted into ordinary differential equations by using a suitable similarity transformation. The analytical solution, of the system of equations is solved by using homotopy analysis method. The convergence region of the obtained solution is determined and plotted. The effects of rotational parameter, porosity of the medium, the characteristics of the non-Newtonian fluid and the suction or injection velocity on the velocity distributions is shown by graphical representation.
Annals of West University of Timisoara - Mathematics and Computer Science, 2018
Bearing in mind the considerable importance of fuzzy differential equations (FDEs) in different f... more Bearing in mind the considerable importance of fuzzy differential equations (FDEs) in different fields of science and engineering, in this paper, nonlinear nth order FDEs are approximated, heuristically. The analysis is carried out on using Chebyshev neural network (ChNN), which is a type of single layer functional link artificial neural network (FLANN). Besides, explication of generalized Hukuhara differentiability (gH-differentiability) is also added for the nth order differentiability of fuzzy-valued functions. Moreover, general formulation of the structure of ChNN for the governing problem is described and assessed on some examples of nonlinear FDEs. In addition, comparison analysis of the proposed method with Runge-Kutta method is added and also portrayed the error bars that clarify the feasibility of attained solutions and validity of the method.
In this paper, an extension is introduced into Max-Min Improved Euler methods for solving initial... more In this paper, an extension is introduced into Max-Min Improved Euler methods for solving initial value problems of fuzzy fractional differential equations (FFDEs). Two modified fractional Euler type methods have been proposed and investigated to obtain numerical solutions of linear and nonlinear FFDEs. The proposed algorithms are tested on various illustrative examples. Exact values are also simulated to compare and discuss the closeness and accuracy of approximations so obtained. Comparatively, tables and graphs results reveal the complete reliability, efficiency and accuracy of the proposed methods.
In this paper the helical flows of fractionalized Maxwell fluid model, through a circular cylinde... more In this paper the helical flows of fractionalized Maxwell fluid model, through a circular cylinder, is studied. The motion is produced by the cylinder that at the initial moment begins to rotate around its axis with an angular velocity Omegatp, and to slide along the same axis with linear velocity Utp. The solutions that have been obtained using Laplace and finite Hankel transforms and presented in series form in terms of the newly defined special function M(z), satisfy all imposed initial and boundary conditions. Moreover, the corresponding solutions for ordinary Maxwell and Newtonian fluid obtained as special cases of the present general solution. Finally, the influence of various pertinent parameters on fluid motion as well as the comparison among different fluids models is analyzed by graphical illustrations.
International Journal of Differential Equations, 2011
We applied a new approach to obtain natural frequency of the nonlinear oscillator with discontinu... more We applied a new approach to obtain natural frequency of the nonlinear oscillator with discontinuity. He's Hamiltonian approach is modified for nonlinear oscillator with discontinuity for which the elastic force term is proportional to sgn(u). We employed this method for higher-order approximate solution of the nonlinear oscillator equation. This property is used to obtain approximate frequency-amplitude relationship of a nonlinear oscillator with high accuracy. Many numerical results are given to prove the efficiency of the suggested technique.
We applied an approach to obtain the natural frequency of the generalized Duffing oscillatoru¨+u+... more We applied an approach to obtain the natural frequency of the generalized Duffing oscillatoru¨+u+α3u3+α5u5+α7u7+⋯+αnun=0and a nonlinear oscillator with a restoring force which is the function of a noninteger power exponent of deflectionu¨+αu|u|n−1=0. This approach is based on involved parameters, initial conditions, and collocation points. For any arbitrary power ofn, the approximate frequency analysis is carried out between the natural frequency and amplitude. The solution procedure is simple, and the results obtained are valid for the whole solution domain.
Abstract In this article, we study numerical solutions of time-fractional fourth-order partial di... more Abstract In this article, we study numerical solutions of time-fractional fourth-order partial differential equations with variable coefficients by introducing the fractional derivative in the sense of Caputo. We implement reliable series solution techniques namely Adomian decomposition method (ADM) and He’s variational iteration method (HVIM). Some applications are presented to highlight the significant features of these techniques. The comparison shows that the solutions obtained are in good agreement with each other and with their respective exact solutions. Some of these types of differential equations arise practically in the theory of transverse vibrations.
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