Browsing by Author "Arif, Muhammad"

Now showing 1 - 18 of 18

Citation Count: Shah, R...et al. (2019). "A Novel Method for the Analytical Solution of Fractional Zakharov–Kuznetsov Equations", Advances in Difference Equations, Vol. 2019, No.1.
A Novel Method for the Analytical Solution of Fractional Zakharov–Kuznetsov Equations
(Springer, 2019) Shah, Rasool; Khan, Hassan; Baleanu, Dumitru; Kumam, Poom; Arif, Muhammad; 56389
In this article, an efficient analytical technique, called Laplace–Adomian decomposition method, is used to obtain the solution of fractional Zakharov– Kuznetsov equations. The fractional derivatives are described in terms of Caputo sense. The solution of the suggested technique is represented in a series form of Adomian components, which is convergent to the exact solution of the given problems. Furthermore, the results of the present method have shown close relations with the exact approaches of the investigated problems. Illustrative examples are discussed, showing the validity of the current method. The attractive and straightforward procedure of the present method suggests that this method can easily be extended for the solutions of other nonlinear fractional-order partial differential equations. © 2019, The Author(s).
Citation Count: Shah, Rasool...et al. (2020). "A semi-analytical method to solve family of Kuramoto-Sivashinsky equations", Journal of Taibah University For Science, Vol. 14, No. 1, pp. 402-411.
A semi-analytical method to solve family of Kuramoto-Sivashinsky equations
(2020) Shah, Rasool; Khan, Hassan; Baleanu, Dumitru; Kumam, Poom; Arif, Muhammad; 56389
In this article, a semi-analytical technique is implemented to solve Kuramoto-Sivashinsky equations. The present method is the combination of two well-known methods namely Laplace transform method and variational iteration method. This hybrid property of the proposed method reduces the numbers of calculations and materials. The accuracy and applicability of the suggested method is confirmed through illustration examples. The accuracy of the proposed method is described in terms of absolute error. It is investigated through graphs and tables that the Laplace transformation and variational iteration method (LVIM) solutions are in good agreement with the exact solution of the problems. The LVIM solutions are also obtained at different fractional-order of the derivative. It is observed through graphs and tables that the fractional-order solutions are convergent to an integer solution as fractional-orders approaches to an integer-order of the problems. In conclusion, the overall implementation of the present method support the validity of the suggested method. Due to simple, straightforward and accurate implementation, the present method can be extended to other non-linear fractional partial differential equations.
Citation Count: Khan, Hassan;...et.al. "An approximate analytical solution of the Navier–Stokes equations within Caputo operator and Elzaki transform decomposition method", Advances in Difference Equations, Vol.2020, No.1.
An approximate analytical solution of the Navier–Stokes equations within Caputo operator and Elzaki transform decomposition method
(2020) Hajira; Khan, Hassan; Khan, Adnan; Kumam, Poom; Baleanu, Dumitru; Arif, Muhammad; 56389
In this article, a hybrid technique of Elzaki transformation and decomposition method is used to solve the Navier–Stokes equations with a Caputo fractional derivative. The numerical simulations and examples are presented to show the validity of the suggested method. The solutions are determined for the problems of both fractional and integer orders by a simple and straightforward procedure. The obtained results are shown and explained through graphs and tables. It is observed that the derived results are very close to the actual solutions of the problems. The fractional solutions are of special interest and have a strong relation with the solution at the integer order of the problems. The numerical examples in this paper are nonlinear and thus handle its solutions in a sophisticated manner. It is believed that this work will make it easy to study the nonlinear dynamics, arising in different areas of research and innovation. Therefore, the current method can be extended for the solution of other higher-order nonlinear problems.
Citation Count: Ali, I...et al. (2012). "An Approximate-Analytical Solution to Analyze Fractional View of Telegraph Equations", IEEE Access, Vol. 8, pp. 25638-25649.
An Approximate-Analytical Solution to Analyze Fractional View of Telegraph Equations
(Institute of Electrical and Electronics Engineers Inc., 2020) Ali, Izaz; Khan, Hassan; Farooq, Umar; Baleanu, Dumitru; Arif, Muhammad; 56389
In the present research article, a modified analytical method is applied to solve time-fractional telegraph equations. The Caputo-operator is used to express the derivative of fractional-order. The present method is the combination of two well-known methods namely Mohan transformation method and Adomian decomposition method. The validity of the proposed technique is confirmed through illustrative examples. It is observed that the obtained solutions have strong contact with the exact solution of the examples. Moreover, it is investigated that the present method has the desired degree of accuracy and provided the graphs closed form solutions of all targeted examples. The graphs have verified the convergence analysis of fractional-order solutions to integer-order solution. In conclusion, the suggested method is simple, straightforward and an effective technique to solve fractional-order partial differential equations.
Citation Count: Khan, Hassan...et al. (2019). "An Efficient Analytical Technique, for The Solution of Fractional-Order Telegraph Equations", Mathematics, Vol. 7, No. 5.
An Efficient Analytical Technique, for The Solution of Fractional-Order Telegraph Equations
(MDPI, 2019) Khan, Hassan; Shah, Rasool; Kumam, Poom; Arif, Muhammad; Baleanu, Dumitru; 56389
In the present article, fractional-order telegraph equations are solved by using the Laplace-Adomian decomposition method. The Caputo operator is used to define the fractional derivative. Series form solutions are obtained for fractional-order telegraph equations by using the proposed method. Some numerical examples are presented to understand the procedure of the Laplace-Adomian decomposition method. As the Laplace-Adomian decomposition procedure has shown the least volume of calculations and high rate of convergence compared to other analytical techniques, the Laplace-Adomian decomposition method is considered to be one of the best analytical techniques for solving fractional-order, non-linear partial differential equationsparticularly the fractional-order telegraph equation.
Citation Count: Khan, Hassan...et al. (2019). "Analytical Solution of Fractional-Order Hyperbolic Telegraph Equation, Using Natural Transform Decomposition Method", Electronics, Vol. 8, No. 9.
Analytical Solution of Fractional-Order Hyperbolic Telegraph Equation, Using Natural Transform Decomposition Method
(MDPI, 2019) Khan, Hassan; Shah, Rasool; Baleanu, Dumitru; Kumam, Poom; Arif, Muhammad; 56389
In the current paper, fractional-order hyperbolic telegraph equations are considered for analytical solutions, using the decomposition method based on natural transformation. The fractional derivative is defined by the Caputo operator. The present technique is implemented for both fractional- and integer-order equations, showing that the current technique is an accurate analytical instrument for the solution of partial differential equations of fractional-order arising in all branches of applied sciences. For this purpose, several examples related to hyperbolic telegraph models are presented to explain the procedure of the suggested method. It is noted that the procedure of the present technique is simple, straightforward, accurate, and found to be a better mathematical technique to solve non-linear fractional partial differential equations.
Citation Count: Khan, H...et al. (2020). ,"Analytical Solutions of (2+Time Fractional Order) Dimensional Physical Models, Using Modified Decomposition Method",Applied Sciences (Switzerland), Vol. 10. No. 1.
Analytical Solutions of (2+Time Fractional Order) Dimensional Physical Models, Using Modified Decomposition Method
(MDPI AG, 2020) Khan, Hassan; Farooq, Umar; Shah, Rasool; Baleanu, Dumitru; Kumam, Poom; Arif, Muhammad; 56389
In this article, a new analytical technique based on an innovative transformation is used to solve (2+time fractional-order) dimensional physical models. The proposed method is the hybrid methodology of Shehu transformation along with Adomian decomposition method. The series form solution is obtained by using the suggested method which provides the desired rate of convergence. Some numerical examples are solved by using the proposed method. The solutions of the targeted problems are represented by graphs which have confirmed closed contact between the exact and obtained solutions of the problems. Based on the novelty and straightforward implementation of the method, it is considered to be one of the best analytical techniques to solve linear and non-linear fractional partial differential equations.
Citation Count: Khan, Hassan...et al. (2020). "Approximate analytical fractional view of convection-diffusion equations", Open Physics, Vol. 18, No. 1, pp. 897-905.
Approximate analytical fractional view of convection-diffusion equations
(2020) Khan, Hassan; Mustafa, Saima; Ali, Izaz; Kumam, Poom; Baleanu, Dumitru; Arif, Muhammad; 56389
In this article, a modified variational iteration method along with Laplace transformation is used for obtaining the solution of fractional-order nonlinear convection-diffusion equations (CDEs). The proposed technique is applied for the first time to solve fractional-order nonlinear CDEs and attain a series-form solution with the quick rate of convergence. Tabular and graphical representations are presented to confirm the reliability of the suggested technique. The solutions are calculated for fractional as well as for integer orders of the problems. The solution graphs of the solutions at various fractional derivatives are plotted. The accuracy is measured in terms of absolute error. The higher degree of accuracy is observed from the table and figures. It is further investigated that fractional solutions have the convergence behavior toward the solution at integer order. The applicability of the present technique is verified by illustrative examples. The simple and effective procedure of the current technique supports its implementation to solve other nonlinear fractional problems in different areas of applied science.
Citation Count: Ali, S...et al. (2019). "Computation of Iterative Solutions Along With Stability Analysis to A Coupled System of Fractional Order Differential Equations",Advances in Difference Equations, Vol. 2019, No. 1.
Computation of Iterative Solutions Along With Stability Analysis to A Coupled System of Fractional Order Differential Equations
(Springer International Publishing, 2019) Ali, Sajjad; Abdeljawad, Thabet; Shah, Kamal; Fahd Jarad; Arif, Muhammad; 234808
In this research article, we investigate sufficient results for the existence, uniqueness and stability analysis of iterative solutions to a coupled system of the nonlinear fractional differential equations (FDEs) with highier order boundary conditions. The foundation of these sufficient techniques is a combination of the scheme of lower and upper solutions together with the method of monotone iterative technique. With the help of the proposed procedure, the convergence criteria for extremal solutions are smoothly achieved. Furthermore, a major aspect is devoted to the investigation of Ulam–Hyers type stability analysis which is also established. For the verification of our work, we provide some suitable examples along with their graphical represntation and errors estimates.
Citation Count: Hajira...et al (2020). "Exact solutions of the Laplace fractional boundary value problems via natural decomposition method", Open Physics, Vol. 18, No. 1, pp. 1178-1187.
Exact solutions of the Laplace fractional boundary value problems via natural decomposition method
(2020) Hajira; Khan, Hassan; Chu, Yu-Ming; Shah, Rasool; Baleanu, Dumitru; Arif, Muhammad; 56389
In this article, exact solutions of some Laplace-type fractional boundary value problems (FBVPs) are investigated via natural decomposition method. The fractional derivatives are described within Caputo operator. The natural decomposition technique is applied for the first time to boundary value problems (BVPs) and found to be an excellent tool to solve the suggested problems. The graphical representation of the exact and derived results is presented to show the reliability of the suggested technique. The present study is mainly concerned with the approximate analytical solutions of some FBVPs. Moreover, the solution graphs have shown that the actual and approximate solutions are very closed to each other. The comparison of the proposed and variational iteration methods is done for integer-order problems. The comparison, support strong relationship between the results of the suggested techniques. The overall analysis and the results obtained have confirmed the effectiveness and the simple procedure of natural decomposition technique for obtaining the solution of BVPs. © 2020 Hajira et al., published by De Gruyter 2020.
Citation Count: Ali, Izaz...et al. (2020). "Fractional View Analysis of Acoustic Wave Equations, Using Fractional-Order Differential Equations", Applied Sciences-Basel, Vol. 10, No. 2.
Fractional View Analysis of Acoustic Wave Equations, Using Fractional-Order Differential Equations
(2020) Ali, Izaz; Khan, Hassan; Shah, Rasool; Baleanu, Dumitru; Kumam, Poom; Arif, Muhammad; 56389
In the present research work, a newly developed technique which is known as variational homotopy perturbation transform method is implemented to solve fractional-order acoustic wave equations. The basic idea behind the present research work is to extend the variational homotopy perturbation method to variational homotopy perturbation transform method. The proposed scheme has confirmed, that it is an accurate and straightforward technique to solve fractional-order partial differential equations. The validity of the method is verified with the help of some illustrative examples. The obtained solutions have shown close contact with the exact solutions. Furthermore, the highest degree of accuracy has been achieved by the suggested method. In fact, the present method can be considered as one of the best analytical techniques compared to other analytical techniques to solve non-linear fractional partial differential equations.
Citation Count: Shah, Rasool...et al. (2020). "Fractional View Analysis of Third Order Kortewege-De Vries Equations, Using a New Analytical Technique", Frontiers in Physics, Vol. 7.
Fractional View Analysis of Third Order Kortewege-De Vries Equations, Using a New Analytical Technique
(2020) Shah, Rasool; Farooq, Umar; Khan, Hassan; Baleanu, Dumitru; Kumam, Poom; Arif, Muhammad; 56389
In the present article, fractional view of third order Kortewege-De Vries equations is presented by a sophisticated analytical technique called Mohand decomposition method. The Caputo fractional derivative operator is used to express fractional derivatives, containing in the targeted problems. Some numerical examples are presented to show the effectiveness of the method for both fractional and integer order problems. From the table, it is investigated that the proposed method has the same rate of convergence as compare to homotopy perturbation transform method. The solution graphs have confirmed the best agreement with the exact solutions of the problems and also revealed that if the sequence of fractional-orders is approaches to integer order, then the fractional order solutions of the problems are converge to an integer order solution. Moreover, the proposed method is straight forward and easy to implement and therefore can be used for other non-linear fractional-order partial differential equations.
Citation Count: Khan, Hassan...et al. (2020). "Laplace decomposition for solving nonlinear system of fractional order partial differential equations", Advances in Difference Equations, Vol. 2020, No. 1.
Laplace decomposition for solving nonlinear system of fractional order partial differential equations
(2020) Khan, Hassan; Shah, Rasool; Kumam, Poom; Baleanu, Dumitru; Arif, Muhammad; 56389
In the present article a modified decomposition method is implemented to solve systems of partial differential equations of fractional-order derivatives. The derivatives of fractional-order are expressed in terms of Caputo operator. The validity of the proposed method is analyzed through illustrative examples. The solution graphs have shown a close contact between the exact and LADM solutions. It is observed that the solutions of fractional-order problems converge towards the solution of an integer-order problem, which confirmed the reliability of the suggested technique. Due to better accuracy and straightforward implementation, the extension of the present method can be made to solve other fractional-order problems.
Citation Count: Khan, Hassan;...et.al. (2020). "Laplace decomposition for solving nonlinear system of fractional order partial differential equations", Advances in Difference Equations, Vol.2020, No.1.
Laplace decomposition for solving nonlinear system of fractional order partial differential equations
(2020) Khan, Hassan; Shah, Rasool; Kumam, Poom; Baleanu, Dumitru; Arif, Muhammad; 56389
In the present article a modified decomposition method is implemented to solve systems of partial differential equations of fractional-order derivatives. The derivatives of fractional-order are expressed in terms of Caputo operator. The validity of the proposed method is analyzed through illustrative examples. The solution graphs have shown a close contact between the exact and LADM solutions. It is observed that the solutions of fractional-order problems converge towards the solution of an integer-order problem, which confirmed the reliability of the suggested technique. Due to better accuracy and straightforward implementation, the extension of the present method can be made to solve other fractional-order problems.
Citation Count: Shah, Rasool...et al. (2019). "Natural Transform Decomposition Method for Solving Fractional-Order Partial Differential Equations with Proportional Delay", Mathematics, Vol. 7, No.6.
Natural Transform Decomposition Method for Solving Fractional-Order Partial Differential Equations with Proportional Delay
(MDPI, 2019) Shah, Rasool; Khan, Hassan; Kumam, Poom; Arif, Muhammad; Baleanu, Dumitru; 56389
In the present article, fractional-order partial differential equations with proportional delay, including generalized Burger equations with proportional delay are solved by using Natural transform decomposition method. Natural transform decomposition method solutions for both fractional and integer orders are obtained in series form, showing higher convergence of the proposed method. Illustrative examples are considered to confirm the validity of the present method. Therefore, Natural transform decomposition method is considered to be one of the best analytical technique, to solve fractional-order linear and non-linear Partial deferential equations particularly fractional-order partial differential equations with proportional delay.
Citation Count: Farooq, Umar...et al. (2019). "Numerical solutions of fractional delay differential equations using Chebyshev wavelet method", Computational & Applied Mathematics, Vol. 38, No. 4.
Numerical solutions of fractional delay differential equations using Chebyshev wavelet method
(Springer Heidelberg, 2019) Farooq, Umar; Khan, Hassan; Baleanu, Dumitru; Arif, Muhammad; 56389
In the present research article, we used a new numerical technique called Chebyshev wavelet method for the numerical solutions of fractional delay differential equations. The Caputo operator is used to define fractional derivatives. The numerical results illustrate the accuracy and reliability of the proposed method. Some numerical examples presented which have shown that the computational study completely supports the compatibility of the suggested method. Similarly, a proposed algorithm can also be applied for other physical problems.
Citation Count: Ali, Nigar...et al. (2017) Study of a class of arbitrary order differential equations by a coincidence degree method, Boundary Value Problems
Study of a class of arbitrary order differential equations by a coincidence degree method
(Springer Open, 2017) Ali, Nigar; Shah, Kamal; Baleanu, Dumitru; Arif, Muhammad; Khan, Rahmat Ali; 56389
In this manuscript, we investigate some appropriate conditions which ensure the existence of at least one solution to a class of fractional order differential equations (FDEs) provided by {-(C)D(q)z(t) = theta(t,z(t)); t is an element of J = [0, 1], q is an element of (1, 2], z(t)vertical bar(t=theta) = phi(z), z(1) = delta(C)D(p)z(eta), p,eta is an element of(0, 1). The nonlinear function theta : J x R -> R is continuous. Further, delta is an element of(0, 1) and phi is an element of C(J, R) is a non-local function. We establish some adequate conditions for the existence of at least one solution to the considered problem by using Gronwall's inequality and a priori estimate tools called the topological degree method. We provide two examples to verify the obtained results.
Citation Count: Shah, Rasool...et al. (2020). "The analytical investigation of time-fractional multi-dimensional Navier–Stokes equation", Alexandria Engineering Journal, Vol. 59, No. 5, pp. 2941-2956.
The analytical investigation of time-fractional multi-dimensional Navier–Stokes equation
(2020) Shah, Rasool; Khan, Hassan; Baleanu, Dumitru; Kumam, Poom; Arif, Muhammad; 56389
In the present research article, we implemented two well-known analytical techniques to solve fractional-order multi-dimensional Navier–Stokes equation. The proposed methods are the modification of Adomian decomposition method and variational iteration method by using natural transformation. Furthermore, some illustrative examples are presented to confirm the validity of the suggested methods. The solutions graphs and tables are constructed for both fractional and integer-order problems. It is investigated that the suggested techniques have the identical solutions of the problems. The solution comparison via graphs and tables have also supported the greater accuracy and higher rate of convergence of the present methods.