Journal of the Nigerian Society of Physical Sciences
This research derives the shifted Jacobi operational matrix (JOM) with respect to fractional deri... more This research derives the shifted Jacobi operational matrix (JOM) with respect to fractional derivatives, implemented with the spectral tau method for the numerical solution of the Atangana-Baleanu Caputo (ABC) derivative. The major aspect of this method is that it considerably simplifies problems by reducing them to ones that can be solved by solving a set of algebraic equations. The main advantage of this method is its high robustness and accuracy gained by a small number of Jacobi functions. The suggested approaches are applied in solving non-linear and linear ABC problems according to initial conditions, and the efficiency and applicability of the proposed method are proved by several test examples. A lot of focus is placed on contrasting the numerical outcomes discovered by the new algorithm together with those discovered by previously well-known methods.
In this article we propose a hybrid method based on a local meshless method and the Laplace trans... more In this article we propose a hybrid method based on a local meshless method and the Laplace transform for approximating the solution of linear one dimensional partial differential equations in the sense of the Caputo–Fabrizio fractional derivative. In our numerical scheme the Laplace transform is used to avoid the time stepping procedure, and the local meshless method is used to produce sparse differentiation matrices and avoid the ill conditioning issues resulting in global meshless methods. Our numerical method comprises three steps. In the first step we transform the given equation to an equivalent time independent equation. Secondly the reduced equation is solved via a local meshless method. Finally, the solution of the original equation is obtained via the inverse Laplace transform by representing it as a contour integral in the complex left half plane. The contour integral is then approximated using the trapezoidal rule. The stability and convergence of the method are discusse...
In this paper, we consider the Space-Time Fractional Advection-Dispersion equation on a finite do... more In this paper, we consider the Space-Time Fractional Advection-Dispersion equation on a finite domain with variable coefficients. Fractional Advection- Dispersion equation as a model for transporting heterogeneous subsurface media as one approach to the modeling of the generally non-Fickian behavior of transport. We use a semi-analytical method as Reproducing kernel Method to solve the Space-Time Fractional Advection-Dispersion equation so that we can get better approximate solutions than the methods with which this problem has been solved. The main obstacle to solve this problem is the existence of a Gram-Schmidt orthogonalization process in the general form of the reproducing kernel method, which is very time-consuming. So, we introduce the Improved Reproducing Kernel Method, which is a different implementation for the general form of the reproducing kernel method. In this method, the Gram-Schmidt orthogonalization process is eliminated to significantly reduce the CPU-time. Also, ...
Abstract Investigated in the present paper is a fifth-order nonlinear evolution (FONLE) equation,... more Abstract Investigated in the present paper is a fifth-order nonlinear evolution (FONLE) equation, known as a nonlinear water wave (NLWW) equation, with applications in the applied sciences. More precisely, a traveling wave hypothesis is firstly applied that reduces the FONLE equation to a 1D domain. The Kudryashov methods (KMs) are then adopted as leading techniques to construct specific wave structures of the governing model which are classified as W -shaped and other solitons. In the end, the effect of changing the coefficients of nonlinear terms on the dynamical features of W -shaped and other solitons is investigated in detail for diverse groups of the involved parameters.
In the present work, the generalized complex Ginzburg–Landau (GCGL) model is considered and its 1... more In the present work, the generalized complex Ginzburg–Landau (GCGL) model is considered and its 1-soliton solutions involving different wave structures are retrieved through a series of newly well-organized methods. More exactly, after considering the GCGL model, its 1-soliton solutions are obtained using the exponential and Kudryashov methods in the presence of perturbation effects. As a case study, the effect of various parameter regimes on the dynamics of the dark and bright soliton solutions is analyzed in three- and two-dimensional postures. The validity of all the exact solutions presented in this study has been examined successfully through the use of the symbolic computation system.
Advances in Intelligent Systems and Computing, 2019
In this present article solution and stability analysis of a fuzzy delay differential equation wi... more In this present article solution and stability analysis of a fuzzy delay differential equation with application is presented. For the presence of the uncertainty the uncertainty parameter namely fuzzy number with the corresponding differential equation in time delay model becomes fuzzy delay differential equation (FDDE) model. Using generalized Hukuhara derivative technique the fuzzy delay differential equation transformed to system of two crisp delay differential equations. The fuzzy stability criterion is found for different cases. The results are followed by a real world problem delayed protein degradation model.
AWERProcedia Information Technology and Computer Science, Dec 24, 2012
Homotopy Analysis Method (HAM) and Homotopy Perturbation Method (HPM) is known to be two powerful... more Homotopy Analysis Method (HAM) and Homotopy Perturbation Method (HPM) is known to be two powerful tools for solving many functional equations such as ordinary and partial differential and integral equations. In this paper (HAM) is applied to solve linear Fredholm and Volterra first and second kind integral equations, the deformation equations are solved analytically by using MATLAB integration functions. Numerical techniques for solving deformation equation are also applied using interpolation methods and Gaussian ...
AWERProcedia Information Technology and Computer Science, Dec 25, 2012
In this paper, it is shown that the new fuzzy house prices can exceed direct fuzzy development co... more In this paper, it is shown that the new fuzzy house prices can exceed direct fuzzy development costs by significant margins in competitive fuzzy housing market. Consequently, some well-known problems have been considered under fuzzy concepts like as fuzzy equilibrium land prices problem, the fuzzy social planner's problem and the fuzzy market value of developed land. Also, some theoretical discussions for the mentioned problems in the fuzzy framework are provided.
In this paper, a novel coronavirus infection system with a fuzzy fractional differential equation... more In this paper, a novel coronavirus infection system with a fuzzy fractional differential equation defined in Caputo’s sense is developed. By using the fuzzy Laplace method coupled with Adomian decomposition transform, numerical results are obtained for better understanding of the dynamical structures of the physical behavior of COVID-19. Such behavior on the general properties of RNA in COVID-19 is also investigated for the governing model. The results demonstrate the efficiency of the proposed approach to address the uncertainty condition in the pandemic situation.
Discrete & Continuous Dynamical Systems - S, 2022
In this research article, the techniques for computing an analytical solution of 2D fuzzy wave eq... more In this research article, the techniques for computing an analytical solution of 2D fuzzy wave equation with some affecting term of force has been provided. Such type of achievement for the aforesaid solution is obtained by applying the notions of a Caputo non-integer derivative in the vague or uncertainty form. At the first attempt the fuzzy natural transform is applied for obtaining the series solution. Secondly the homotopy perturbation (HPM) technique is used, for the analysis of the proposed result by comparing the co-efficient of homotopy parameter \begin{document}$ q $\end{document} to get hierarchy of equation of different order for \begin{document}$ q $\end{document}. For this purpose, some new results about Natural transform of an arbitrary derivative under uncertainty are established, for the first time in the literature. The solution has been assumed in term of infinite series, which break the problem to a small number of equations, for the respective investigation. The ...
Now-a-days, uncertainty conditions play an important role in modelling of real-world problems. In... more Now-a-days, uncertainty conditions play an important role in modelling of real-world problems. In this regard, the aim of this study is two folded. Firstly, the concept of system of interval differential equations and its solution procedure in the parametric approach have been proposed. To serve this purpose, using parametric representation of interval and its arithmetic, system of linear interval differential equations is converted to the system of differential equations in parametric form. Then, a mixing problem with three liquids is considered and the mixing process is governed by system of interval differential equations. Thereafter, the mixing liquid is used in the production process of a manufacturing firm. Secondly, using this concept, a production inventory model for single item has been developed by employing mixture of liquids and the proposed production system is formulated mathematically by using system of interval differential equations.The corresponding interval valued...
Journal of the Nigerian Society of Physical Sciences
This research derives the shifted Jacobi operational matrix (JOM) with respect to fractional deri... more This research derives the shifted Jacobi operational matrix (JOM) with respect to fractional derivatives, implemented with the spectral tau method for the numerical solution of the Atangana-Baleanu Caputo (ABC) derivative. The major aspect of this method is that it considerably simplifies problems by reducing them to ones that can be solved by solving a set of algebraic equations. The main advantage of this method is its high robustness and accuracy gained by a small number of Jacobi functions. The suggested approaches are applied in solving non-linear and linear ABC problems according to initial conditions, and the efficiency and applicability of the proposed method are proved by several test examples. A lot of focus is placed on contrasting the numerical outcomes discovered by the new algorithm together with those discovered by previously well-known methods.
In this article we propose a hybrid method based on a local meshless method and the Laplace trans... more In this article we propose a hybrid method based on a local meshless method and the Laplace transform for approximating the solution of linear one dimensional partial differential equations in the sense of the Caputo–Fabrizio fractional derivative. In our numerical scheme the Laplace transform is used to avoid the time stepping procedure, and the local meshless method is used to produce sparse differentiation matrices and avoid the ill conditioning issues resulting in global meshless methods. Our numerical method comprises three steps. In the first step we transform the given equation to an equivalent time independent equation. Secondly the reduced equation is solved via a local meshless method. Finally, the solution of the original equation is obtained via the inverse Laplace transform by representing it as a contour integral in the complex left half plane. The contour integral is then approximated using the trapezoidal rule. The stability and convergence of the method are discusse...
In this paper, we consider the Space-Time Fractional Advection-Dispersion equation on a finite do... more In this paper, we consider the Space-Time Fractional Advection-Dispersion equation on a finite domain with variable coefficients. Fractional Advection- Dispersion equation as a model for transporting heterogeneous subsurface media as one approach to the modeling of the generally non-Fickian behavior of transport. We use a semi-analytical method as Reproducing kernel Method to solve the Space-Time Fractional Advection-Dispersion equation so that we can get better approximate solutions than the methods with which this problem has been solved. The main obstacle to solve this problem is the existence of a Gram-Schmidt orthogonalization process in the general form of the reproducing kernel method, which is very time-consuming. So, we introduce the Improved Reproducing Kernel Method, which is a different implementation for the general form of the reproducing kernel method. In this method, the Gram-Schmidt orthogonalization process is eliminated to significantly reduce the CPU-time. Also, ...
Abstract Investigated in the present paper is a fifth-order nonlinear evolution (FONLE) equation,... more Abstract Investigated in the present paper is a fifth-order nonlinear evolution (FONLE) equation, known as a nonlinear water wave (NLWW) equation, with applications in the applied sciences. More precisely, a traveling wave hypothesis is firstly applied that reduces the FONLE equation to a 1D domain. The Kudryashov methods (KMs) are then adopted as leading techniques to construct specific wave structures of the governing model which are classified as W -shaped and other solitons. In the end, the effect of changing the coefficients of nonlinear terms on the dynamical features of W -shaped and other solitons is investigated in detail for diverse groups of the involved parameters.
In the present work, the generalized complex Ginzburg–Landau (GCGL) model is considered and its 1... more In the present work, the generalized complex Ginzburg–Landau (GCGL) model is considered and its 1-soliton solutions involving different wave structures are retrieved through a series of newly well-organized methods. More exactly, after considering the GCGL model, its 1-soliton solutions are obtained using the exponential and Kudryashov methods in the presence of perturbation effects. As a case study, the effect of various parameter regimes on the dynamics of the dark and bright soliton solutions is analyzed in three- and two-dimensional postures. The validity of all the exact solutions presented in this study has been examined successfully through the use of the symbolic computation system.
Advances in Intelligent Systems and Computing, 2019
In this present article solution and stability analysis of a fuzzy delay differential equation wi... more In this present article solution and stability analysis of a fuzzy delay differential equation with application is presented. For the presence of the uncertainty the uncertainty parameter namely fuzzy number with the corresponding differential equation in time delay model becomes fuzzy delay differential equation (FDDE) model. Using generalized Hukuhara derivative technique the fuzzy delay differential equation transformed to system of two crisp delay differential equations. The fuzzy stability criterion is found for different cases. The results are followed by a real world problem delayed protein degradation model.
AWERProcedia Information Technology and Computer Science, Dec 24, 2012
Homotopy Analysis Method (HAM) and Homotopy Perturbation Method (HPM) is known to be two powerful... more Homotopy Analysis Method (HAM) and Homotopy Perturbation Method (HPM) is known to be two powerful tools for solving many functional equations such as ordinary and partial differential and integral equations. In this paper (HAM) is applied to solve linear Fredholm and Volterra first and second kind integral equations, the deformation equations are solved analytically by using MATLAB integration functions. Numerical techniques for solving deformation equation are also applied using interpolation methods and Gaussian ...
AWERProcedia Information Technology and Computer Science, Dec 25, 2012
In this paper, it is shown that the new fuzzy house prices can exceed direct fuzzy development co... more In this paper, it is shown that the new fuzzy house prices can exceed direct fuzzy development costs by significant margins in competitive fuzzy housing market. Consequently, some well-known problems have been considered under fuzzy concepts like as fuzzy equilibrium land prices problem, the fuzzy social planner's problem and the fuzzy market value of developed land. Also, some theoretical discussions for the mentioned problems in the fuzzy framework are provided.
In this paper, a novel coronavirus infection system with a fuzzy fractional differential equation... more In this paper, a novel coronavirus infection system with a fuzzy fractional differential equation defined in Caputo’s sense is developed. By using the fuzzy Laplace method coupled with Adomian decomposition transform, numerical results are obtained for better understanding of the dynamical structures of the physical behavior of COVID-19. Such behavior on the general properties of RNA in COVID-19 is also investigated for the governing model. The results demonstrate the efficiency of the proposed approach to address the uncertainty condition in the pandemic situation.
Discrete & Continuous Dynamical Systems - S, 2022
In this research article, the techniques for computing an analytical solution of 2D fuzzy wave eq... more In this research article, the techniques for computing an analytical solution of 2D fuzzy wave equation with some affecting term of force has been provided. Such type of achievement for the aforesaid solution is obtained by applying the notions of a Caputo non-integer derivative in the vague or uncertainty form. At the first attempt the fuzzy natural transform is applied for obtaining the series solution. Secondly the homotopy perturbation (HPM) technique is used, for the analysis of the proposed result by comparing the co-efficient of homotopy parameter \begin{document}$ q $\end{document} to get hierarchy of equation of different order for \begin{document}$ q $\end{document}. For this purpose, some new results about Natural transform of an arbitrary derivative under uncertainty are established, for the first time in the literature. The solution has been assumed in term of infinite series, which break the problem to a small number of equations, for the respective investigation. The ...
Now-a-days, uncertainty conditions play an important role in modelling of real-world problems. In... more Now-a-days, uncertainty conditions play an important role in modelling of real-world problems. In this regard, the aim of this study is two folded. Firstly, the concept of system of interval differential equations and its solution procedure in the parametric approach have been proposed. To serve this purpose, using parametric representation of interval and its arithmetic, system of linear interval differential equations is converted to the system of differential equations in parametric form. Then, a mixing problem with three liquids is considered and the mixing process is governed by system of interval differential equations. Thereafter, the mixing liquid is used in the production process of a manufacturing firm. Secondly, using this concept, a production inventory model for single item has been developed by employing mixture of liquids and the proposed production system is formulated mathematically by using system of interval differential equations.The corresponding interval valued...
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