European Journal of Pure and Applied Mathematics, 2023
Recently, Shehu has introduced an integral transform called Shehu transform, which generalizes th... more Recently, Shehu has introduced an integral transform called Shehu transform, which generalizes the two well-known integrals transforms, i.e. Laplace and Sumudu transform. In the literature, many integral transforms were used to compute the solution of integro-differential equations (IDEs). In this article, for the first time, we use Shehu transform for the computation of solution of n th-order IDEs. We present a general scheme of solution for n th-order IDEs. We give some examples with detailed solutions to show the appropriateness of the method. We present the accuracy, simplicity, and convergence of the proposed method through tables and graphs.
The aim of the present analysis is to apply Adomian decomposition method for the solution of a ti... more The aim of the present analysis is to apply Adomian decomposition method for the solution of a time-fractional Navier–Stokes equation in a tube. By using an initial value, the explicit solution of the equation has been presented in the closed form and then its numerical solution has been represented graphically. The present method performs extremely well in terms of efficiency
In this paper, a new analytical method to linear and nonlinear partial differential equation call... more In this paper, a new analytical method to linear and nonlinear partial differential equation called the Natural Homotopy Perturbation Method (NHPM) is introduced. The new analytical method is a combination of the Natural Transform Method (NTM) and a well known analytical method, Homotopy Perturbation Method (HPM). The proposed analytical method avoids round off errors, linearization, transformation or taking some restrictive assumptions. Exact solution of linear and nonlinear partial differential equation are successfully obtained using the new analytical method, and the results are compared with the results of the existing methods. The high simplicity, efficiency, and accuracy of the new analytical method are clearly demonstrated.
In this paper, we propose the fuzzy Shehu transform method (FSTM) using Zadeh’s decomposition the... more In this paper, we propose the fuzzy Shehu transform method (FSTM) using Zadeh’s decomposition theorem and fuzzy Riemann integral of real-valued functions on finite intervals. As an alternative to standard fuzzy Laplace transform and the fuzzy Sumudu integral transform, we established some potential useful (new or known) properties of the FSTM and validate their applications. Furthermore, the FSTM is coupled with the well-known homotopy analysis method to obtain the approximate and exact solutions of fuzzy differential equations of integer and non-integer order derivatives. The convergence analysis and the error analysis of the suggested technique are provided and supported by graphical solutions. Comparison of the numerical simulations of exact and approximate solutions of two fuzzy fractional partial differential equations are tabulated to further justify the reliability and efficiency of the proposed method.
Progress in Fractional Differentiation and Applications, 2018
A new analytical method called the local fractional natural homotopy perturbation method (LFNHPM)... more A new analytical method called the local fractional natural homotopy perturbation method (LFNHPM) for solving partial differential equations with local fractional derivative is introduced. The proposed analytical method is a combination of the local fractional homotopy perturbation method (LFHPM) and the local fractional natural transform (LFNTM). In this analytical method, the fractional derivative operators are computed in local fractional sense, and the nonlinear terms are calculated using He's polynomial. Some applications are given to illustrate the simplicity, efficiency, and high accuracy of the proposed method. Keywords: Local fractional natural homotopy perturbation method, local fractional derivative operator, local fractional partial differential equations.
International Journal of Differential Equations, 2016
A hybrid analytical method for solving linear and nonlinear fractional partial differential equat... more A hybrid analytical method for solving linear and nonlinear fractional partial differential equations is presented. The proposed analytical approach is an elegant combination of the Natural Transform Method (NTM) and a well-known method, Homotopy Perturbation Method (HPM). In this analytical method, the fractional derivative is computed in Caputo sense and the nonlinear term is calculated using He’s polynomial. The proposed analytical method reduces the computational size and avoids round-off errors. Exact solution of linear and nonlinear fractional partial differential equations is successfully obtained using the analytical method.
Progress in Fractional Differentiation and Applications, 2016
In this paper, a new analytical method called the Natural Homotopy Perturbation Method (NHPM) for... more In this paper, a new analytical method called the Natural Homotopy Perturbation Method (NHPM) for solving linear and the nonlinear fractional partial differential equation is introduced. The proposed analytical method is an elegant combination of a well-known method, Homotopy Perturbation Method (HPM) and the Natural Transform Method (NTM). In this new analytical method, the fractional derivative is computed in Caputo sense and the nonlinear terms are calculated using He's polynomials. Exact solution of linear and nonlinear fractional partial differential equations are successfully obtained using the new analytical method, and the result is compared with the result of the existing methods.
In this paper, we propose a new analytical modelling of the well-known fractional generalized Kur... more In this paper, we propose a new analytical modelling of the well-known fractional generalized Kuramoto-Sivashinky equation (FGKSE) using fractional operator with non-singular kernel and the homotopy analysis transform method via J-transform method. Also, using fixed-point theorem, we prove the existence and uniqueness of our proposed solution to the fractional Kuramoto-Sivashinky equation. To further validate the efficiency of the suggested technique, we proved the convergence analysis of the method and provide the error estimate. The obtained solutions of the FGKSE, describing turbulence processes in the field of ocean engineering are analytically and numerically compared to show the behaviors of many parameters of the present model.
In this paper, we propose a new analytical modelling of the well-known fractional generalized Kur... more In this paper, we propose a new analytical modelling of the well-known fractional generalized Kuramoto-Sivashinky equation (FGKSE) using fractional operator with non-singular kernel and the homotopy analysis transform method via J-transform method. Also, using fixed-point theorem, we prove the existence and uniqueness of our proposed solution to the fractional Kuramoto-Sivashinky equation. To further validate the efficiency of the suggested technique, we proved the convergence analysis of the method and provide the error estimate. The obtained solutions of the FGKSE, describing turbulence processes in the field of ocean engineering are analytically and numerically compared to show the behaviors of many parameters of the present model.
In this paper, we introduce a semi-analytical method called the local fractional Laplace homotopy... more In this paper, we introduce a semi-analytical method called the local fractional Laplace homotopy analysis method (LFLHAM) for solving wave equations with local fractional derivatives. The LFLHAM is based on the homotopy analysis method and the local fractional Laplace transform method, respectively. The proposed analytical method was a modification of the homotopy analysis method and converged rapidly within a few iterations. The nonzero convergence-control parameter was used to adjust the convergence of the series solutions. Three examples of non-differentiable wave equations were provided to demonstrate the efficiency and the high accuracy of the proposed technique. The results obtained were completely in agreement with the results in the existing methods and their qualitative and quantitative comparison of the results.
In this article, we develop a new method called the Natural Decomposition Method (NDM). We use th... more In this article, we develop a new method called the Natural Decomposition Method (NDM). We use the NDM to find exact solutions for coupled systems of linear and nonlinear partial differential equations. The proposed method is a combination of the Natural Transform Method (NTM) and Adomian Decomposition method (ADM). Besides, the NDM avoid round-off errors which leads to solutions in closed form. The new method always lead to an exact or approximate solution in the form of rapidly convergence series. Hence, the Natural Decomposition Method is elegant refinement of the existing methods and can easily be used to solve a wide class of Linear and Nonlinear Partial Differential Equations.
The fundamental purpose of this paper is to propose a new Laplace-type integral transform (NL-TIT... more The fundamental purpose of this paper is to propose a new Laplace-type integral transform (NL-TIT) for solving steady heat-transfer problems. The proposed integral transform is a generalization of the Sumudu, and the Laplace transforms and its visualization is more comfortable than the Sumudu transform, the natural transform, and the Elzaki transform. The suggested integral transform is used to solve the steady heat-transfer problems, and results are compared with the results of the existing techniques.
European Journal of Pure and Applied Mathematics, 2023
Recently, Shehu has introduced an integral transform called Shehu transform, which generalizes th... more Recently, Shehu has introduced an integral transform called Shehu transform, which generalizes the two well-known integrals transforms, i.e. Laplace and Sumudu transform. In the literature, many integral transforms were used to compute the solution of integro-differential equations (IDEs). In this article, for the first time, we use Shehu transform for the computation of solution of n th-order IDEs. We present a general scheme of solution for n th-order IDEs. We give some examples with detailed solutions to show the appropriateness of the method. We present the accuracy, simplicity, and convergence of the proposed method through tables and graphs.
The aim of the present analysis is to apply Adomian decomposition method for the solution of a ti... more The aim of the present analysis is to apply Adomian decomposition method for the solution of a time-fractional Navier–Stokes equation in a tube. By using an initial value, the explicit solution of the equation has been presented in the closed form and then its numerical solution has been represented graphically. The present method performs extremely well in terms of efficiency
In this paper, a new analytical method to linear and nonlinear partial differential equation call... more In this paper, a new analytical method to linear and nonlinear partial differential equation called the Natural Homotopy Perturbation Method (NHPM) is introduced. The new analytical method is a combination of the Natural Transform Method (NTM) and a well known analytical method, Homotopy Perturbation Method (HPM). The proposed analytical method avoids round off errors, linearization, transformation or taking some restrictive assumptions. Exact solution of linear and nonlinear partial differential equation are successfully obtained using the new analytical method, and the results are compared with the results of the existing methods. The high simplicity, efficiency, and accuracy of the new analytical method are clearly demonstrated.
In this paper, we propose the fuzzy Shehu transform method (FSTM) using Zadeh’s decomposition the... more In this paper, we propose the fuzzy Shehu transform method (FSTM) using Zadeh’s decomposition theorem and fuzzy Riemann integral of real-valued functions on finite intervals. As an alternative to standard fuzzy Laplace transform and the fuzzy Sumudu integral transform, we established some potential useful (new or known) properties of the FSTM and validate their applications. Furthermore, the FSTM is coupled with the well-known homotopy analysis method to obtain the approximate and exact solutions of fuzzy differential equations of integer and non-integer order derivatives. The convergence analysis and the error analysis of the suggested technique are provided and supported by graphical solutions. Comparison of the numerical simulations of exact and approximate solutions of two fuzzy fractional partial differential equations are tabulated to further justify the reliability and efficiency of the proposed method.
Progress in Fractional Differentiation and Applications, 2018
A new analytical method called the local fractional natural homotopy perturbation method (LFNHPM)... more A new analytical method called the local fractional natural homotopy perturbation method (LFNHPM) for solving partial differential equations with local fractional derivative is introduced. The proposed analytical method is a combination of the local fractional homotopy perturbation method (LFHPM) and the local fractional natural transform (LFNTM). In this analytical method, the fractional derivative operators are computed in local fractional sense, and the nonlinear terms are calculated using He's polynomial. Some applications are given to illustrate the simplicity, efficiency, and high accuracy of the proposed method. Keywords: Local fractional natural homotopy perturbation method, local fractional derivative operator, local fractional partial differential equations.
International Journal of Differential Equations, 2016
A hybrid analytical method for solving linear and nonlinear fractional partial differential equat... more A hybrid analytical method for solving linear and nonlinear fractional partial differential equations is presented. The proposed analytical approach is an elegant combination of the Natural Transform Method (NTM) and a well-known method, Homotopy Perturbation Method (HPM). In this analytical method, the fractional derivative is computed in Caputo sense and the nonlinear term is calculated using He’s polynomial. The proposed analytical method reduces the computational size and avoids round-off errors. Exact solution of linear and nonlinear fractional partial differential equations is successfully obtained using the analytical method.
Progress in Fractional Differentiation and Applications, 2016
In this paper, a new analytical method called the Natural Homotopy Perturbation Method (NHPM) for... more In this paper, a new analytical method called the Natural Homotopy Perturbation Method (NHPM) for solving linear and the nonlinear fractional partial differential equation is introduced. The proposed analytical method is an elegant combination of a well-known method, Homotopy Perturbation Method (HPM) and the Natural Transform Method (NTM). In this new analytical method, the fractional derivative is computed in Caputo sense and the nonlinear terms are calculated using He's polynomials. Exact solution of linear and nonlinear fractional partial differential equations are successfully obtained using the new analytical method, and the result is compared with the result of the existing methods.
In this paper, we propose a new analytical modelling of the well-known fractional generalized Kur... more In this paper, we propose a new analytical modelling of the well-known fractional generalized Kuramoto-Sivashinky equation (FGKSE) using fractional operator with non-singular kernel and the homotopy analysis transform method via J-transform method. Also, using fixed-point theorem, we prove the existence and uniqueness of our proposed solution to the fractional Kuramoto-Sivashinky equation. To further validate the efficiency of the suggested technique, we proved the convergence analysis of the method and provide the error estimate. The obtained solutions of the FGKSE, describing turbulence processes in the field of ocean engineering are analytically and numerically compared to show the behaviors of many parameters of the present model.
In this paper, we propose a new analytical modelling of the well-known fractional generalized Kur... more In this paper, we propose a new analytical modelling of the well-known fractional generalized Kuramoto-Sivashinky equation (FGKSE) using fractional operator with non-singular kernel and the homotopy analysis transform method via J-transform method. Also, using fixed-point theorem, we prove the existence and uniqueness of our proposed solution to the fractional Kuramoto-Sivashinky equation. To further validate the efficiency of the suggested technique, we proved the convergence analysis of the method and provide the error estimate. The obtained solutions of the FGKSE, describing turbulence processes in the field of ocean engineering are analytically and numerically compared to show the behaviors of many parameters of the present model.
In this paper, we introduce a semi-analytical method called the local fractional Laplace homotopy... more In this paper, we introduce a semi-analytical method called the local fractional Laplace homotopy analysis method (LFLHAM) for solving wave equations with local fractional derivatives. The LFLHAM is based on the homotopy analysis method and the local fractional Laplace transform method, respectively. The proposed analytical method was a modification of the homotopy analysis method and converged rapidly within a few iterations. The nonzero convergence-control parameter was used to adjust the convergence of the series solutions. Three examples of non-differentiable wave equations were provided to demonstrate the efficiency and the high accuracy of the proposed technique. The results obtained were completely in agreement with the results in the existing methods and their qualitative and quantitative comparison of the results.
In this article, we develop a new method called the Natural Decomposition Method (NDM). We use th... more In this article, we develop a new method called the Natural Decomposition Method (NDM). We use the NDM to find exact solutions for coupled systems of linear and nonlinear partial differential equations. The proposed method is a combination of the Natural Transform Method (NTM) and Adomian Decomposition method (ADM). Besides, the NDM avoid round-off errors which leads to solutions in closed form. The new method always lead to an exact or approximate solution in the form of rapidly convergence series. Hence, the Natural Decomposition Method is elegant refinement of the existing methods and can easily be used to solve a wide class of Linear and Nonlinear Partial Differential Equations.
The fundamental purpose of this paper is to propose a new Laplace-type integral transform (NL-TIT... more The fundamental purpose of this paper is to propose a new Laplace-type integral transform (NL-TIT) for solving steady heat-transfer problems. The proposed integral transform is a generalization of the Sumudu, and the Laplace transforms and its visualization is more comfortable than the Sumudu transform, the natural transform, and the Elzaki transform. The suggested integral transform is used to solve the steady heat-transfer problems, and results are compared with the results of the existing techniques.
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