Browsing by Author "Khan, Rahmat Ali"

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Citation Count: Khalil, H...et al. (2016). Approximate solution of linear and nonlinear fractional differential equations under m-point local and nonlocal boundary conditions. Advance in Difference Equations. http://dx.doi.org/ 10.1186/s13662-016-0910-7
Approximate solution of linear and nonlinear fractional differential equations under m-point local and nonlocal boundary conditions
(Springer International Publishing, 2016) Khalil, Hammad; Khan, Rahmat Ali; Baleanu, Dumitru; Saker, Samir H.
This paper investigates a computational method to find an approximation to the solution of fractional differential equations subject to local and nonlocal m-point boundary conditions. The method that we will employ is a variant of the spectral method which is based on the normalized Bernstein polynomials and its operational matrices. Operational matrices that we will developed in this paper have the ability to convert fractional differential equations together with its nonlocal boundary conditions to a system of easily solvable algebraic equations. Some test problems are presented to illustrate the efficiency, accuracy, and applicability of the proposed method.
Citation Count: Khan, Rahmat Ali; Li, Yongjin; Jarad, Fahd (2021). "Exact analytical solutions of fractional order telegraph equations via triple laplace transform", Discrete and Continuous Dynamical Systems - Series S, Vol. 14, No. 7, pp. 2387-2397.
Exact analytical solutions of fractional order telegraph equations via triple laplace transform
(2021) Khan, Rahmat Ali; Li, Yongjin; Jarad, Fahd; 234808
In this paper, we study initial/boundary value problems for 1 + 1 dimensional and 1 + 2 dimensional fractional order telegraph equations. We develop the technique of double and triple Laplace transforms and obtain exact analytical solutions of these problems. The techniques we develop are new and are not limited to only telegraph equations but can be used for exact solutions of large class of linear fractional order partial differential equations © 2021 American Institute of Mathematical Sciences. All rights reserved.
Citation Count: Jafari, H...et al. (2015). Existence criterion for the solutions of fractional order p-Laplacian boundary value problems. Boundray Value Problems. http://dx.doi.org/ 10.1186/s13661-015-0425-2
Existence criterion for the solutions of fractional order p-Laplacian boundary value problems
(Springer International Publishing, 2015) Jafari, Hossein; Baleanu, Dumitru; Khan, Hasib; Khan, Rahmat Ali; Khan, Aziz
The existence criterion has been extensively studied for different classes in fractional differential equations (FDEs) through different mathematical methods. The class of fractional order boundary value problems (FOBVPs) with p-Laplacian operator is one of the most popular class of the FDEs which have been recently considered by many scientists as regards the existence and uniqueness. In this scientific work our focus is on the existence and uniqueness of the FOBVP with p-Laplacian operator of the form: D-gamma(phi(p)(D-theta z(t))) + a(t)f(z(t)) = 0, 3 < theta, gamma <= 4, t is an element of [0, 1], z(0) = z'''(0), eta D(alpha)z(t)vertical bar(t=1) = z'(0), xi z ''(1) - z ''(0) = 0, 0 < alpha < 1, phi(p)(D-theta z(t))vertical bar(t=0) = 0 = (phi(p)(D-theta z(t)))'vertical bar(t=0), (phi(p)(D-theta z(t)))''vertical bar(t=1) = 1/2(phi(p)(D-theta z(t)))''vertical bar(t=0), (phi(p)(D-theta z(t)))'''vertical bar(t=0) = 0, where 0 < xi, eta < 1 and D-theta, D-gamma, D-alpha are Caputo's fractional derivatives of orders theta, gamma, alpha, respectively. For this purpose, we apply Schauder's fixed point theorem and the results are checked by illustrative examples
Citation Count: Khan, Rahmat Ali...et al. (2021). "Existence results for a general class of sequential hybrid fractional differential equations", Advances in Difference Equations, Vol. 2021, No. 1.
Existence results for a general class of sequential hybrid fractional differential equations
(2021) Khan, Rahmat Ali; Gul, Shaista; Jarad, Fahd; Khan, Hasib; 234808
In this paper, we study a class of nonlinear boundary value problems (BVPs) consisting of a more general class of sequential hybrid fractional differential equations (SHFDEs) together with a class of nonlinear boundary conditions at both end points of the domain. The nonlinear functions involved depend explicitly on the fractional derivatives. We study the necessary conditions required for the unique solution to the suggested BVP under the Caratheodory conditions using the technique of measure of noncompactness and degree theory. We also develop conditions for uniqueness results and also on stability analysis. © 2021, The Author(s).
Citation Count: Baleanu, D...et al. (2015). On existence results for solutions of a coupled system of hybrid boundary value problems with hybrid conditions. Advance in Difference Equations. http://dx.doi.org/10.1186/s13662-015-0651-z
On existence results for solutions of a coupled system of hybrid boundary value problems with hybrid conditions
(Springer International Publishing, 2015) Baleanu, Dumitru; Khan, Hasib; Jafari, Hossein; Khan, Rahmat Ali; Alipour, Mohsen
We investigate sufficient conditions for existence and uniqueness of solutions for a coupled system of fractional order hybrid differential equations (HDEs) with multi-point hybrid boundary conditions given by D-omega(x(t)/H(t, x(t), z(t))) = -K-1 (t, x(t), z(t)), omega epsilon (2, 3], D-epsilon(z(t)/G(t, x(t), z(t))) = -K-2 (t, x(t), z(t)), epsilon epsilon(2, 3] x(t)/H(t, x(t), z(t))vertical bar(t=1) = 0, D-mu(x(t)/H(t, x(t), z(t)))vertical bar(t=delta 1) =0, x((2))(0) = 0 z(t)/G(t, x(t), z(t))vertical bar(t=1) = 0, D-nu(z(t)/G(t, x(t), z(t)))vertical bar(t=delta 2) =0, z((2))(0) = 0 where t epsilon [0, 1], delta(1), delta(2), mu, upsilon epsilon (0, 1), and D-omega, D-epsilon, D-mu and D-upsilon are Caputo's fractional derivatives of order omega, is an element of, mu and nu, respectively, K-1, K-2 epsilon C([0, 1] x R x R, R) and G, H epsilon C([0, 1] x R x R, R - {0}). We use classical results due to Dhage and Banach's contraction principle (BCP) for the existence and uniqueness of solutions. For applications of our results, we include examples.
Citation Count: Khan, Hasib...et al. (2020). "On Iterative Solutions and Error Estimations of a Coupled System of Fractional Order Differential-Integral Equations with Initial and Boundary Conditions", Differential Equations and Dynamical Systems, Vol. 28, no. 4, pp. 1059-1071.
On Iterative Solutions and Error Estimations of a Coupled System of Fractional Order Differential-Integral Equations with Initial and Boundary Conditions
(2020) Khan, Hasib; Jafari, Hossein; Baleanu, Dumitru; Khan, Rahmat Ali; Khan, Aziz; 56389
The study of boundary value problems (BVPs) for fractional differential–integral equations (FDIEs) is extremely popular in the scientific community. Scientists are utilizing BVPs for FDIEs in day life problems by the help of different approaches. In this paper, we apply monotone iterative technique for the existence, uniqueness and the error estimations of solutions for a coupled system of BVPs for FDIEs of orders ω, ϵ∈ (3 , 4]. The coupled system is given by Dωu(t)=-G1(t,Iωu(t),Iεv(t)),Dεv(t)=-G2(t,Iωu(t),Iεv(t)),Dδu(1)=0=I3-ωu(0)=I4-ωu(0),u(1)=Γ(ω-δ)Γ(ω)Iω-δG1(t,Iωu(t),Iεv(t))(t=1),Dνv(1)=0=I3-εv(0)=I4-νv(0),v(1)=Γ(ε-ν)Γ(ε)Iε-νG2(t,Iωu(t),Iεv(t))(t=1),where t∈ [0 , 1] , δ, ν∈ [1 , 2]. The functions G1, G2: [0 , 1] × R× R→ R, satisfy the Caratheodory conditions. The fractional derivatives Dω, Dε, Dδ, Dν are in Riemann-Liouville sense and Iω, Iε, I3-ω, I4-ω, I3-ε, I4-ε, Iω-δ, Iε-ν are fractional order integrals. The assumed technique is a better approach for the existence, uniqueness and error estimation. The applications of the results are examined by the help of examples. © 2017, Foundation for Scientific Research and Technological Innovation.
Citation Count: Baleanu, D...et al. (2015). On the exact solution of wave equations on cantor sets. Entropy, 17(9), 6229-6237. http://dx.doi.org/10.3390/e17096229
On the exact solution of wave equations on cantor sets
(MDPI AG, 2015) Baleanu, Dumitru; Khan, Hasib; Jafari, Hossein; Khan, Rahmat Ali
The transfer of heat due to the emission of electromagnetic waves is called thermal radiations. In local fractional calculus, there are numerous contributions of scientists, like Mandelbrot, who described fractal geometry and its wide range of applications in many scientific fields. Christianto and Rahul gave the derivation of Proca equations on Cantor sets. Hao et al. investigated the Helmholtz and diffusion equations in Cantorian and Cantor-Type Cylindrical Coordinates. Carpinteri and Sapora studied diffusion problems in fractal media in Cantor sets. Zhang et al. studied local fractional wave equations under fixed entropy. In this paper, we are concerned with the exact solutions of wave equations by the help of local fractional Laplace variation iteration method (LFLVIM). We develop an iterative scheme for the exact solutions of local fractional wave equations (LFWEs). The efficiency of the scheme is examined by two illustrative examples.
Citation Count: Baleanu, D...et al. (2015). On the existence of solution for fractional differential equations of order 3 < delta(1) <= 4. Advance in Difference Equations. http://dx.doi.org/10.1186/s13662-015-0686-1
On the existence of solution for fractional differential equations of order 3 < delta(1) <= 4
(Springer International Publishing, 2015) Baleanu, Dumitru; Agarwal, Ravi P.; Khan, Hasib; Khan, Rahmat Ali; Jafari, Hossein
In this paper, we deal with a fractional differential equation of order delta(1) is an element of (3,4] with initial and boundary conditions, D-delta 1 psi(x) = -H(x,psi(x)), D-alpha 1 psi(1) = 0 = I3-delta 1 psi(0) = I4-delta 1 psi(0), psi(1) = Gamma(delta(1)-alpha(1))/Gamma(nu(1)) I delta 1-alpha 1 H(x,psi(x))(1), where x is an element of [0, 1], alpha(1) is an element of (1, 2], addressing the existence of a positive solution (EPS), where the fractional derivatives D-delta 1, D-alpha 1 are in the Riemann-Liouville sense of the order delta(1), alpha(1), respectively. The function H is an element of C([0, 1] x R, R) and I delta 1-alpha 1 H(x, psi(x))(1) = 1/Gamma(delta(1)-alpha(1)) integral(1)(0) (1 -z)(delta 1-alpha 1-1) H(z,psi(z)) dz. To this aim, we establish an equivalent integral form of the problem with the help of a Green's function. We also investigate the properties of the Green's function in the paper which we utilize in our main result for the EPS of the problem. Results for the existence of solutions are obtained with the help of some classical results
Citation Count: Baleanu, Dumitru;...et.al (2015). "On the existence of solution for fractional differential equations of order 3< δ1≤4", Advances in Difference Equations, Vol.2015, No.1, pp.1-9.
On the existence of solution for fractional differential equations of order 3< δ1≤4
(2015) Baleanu, Dumitru; Agarwal, Ravi P; Khan, Hasib; Khan, Rahmat Ali; Jafari, Hossein; 56389
In this paper, we deal with a fractional differential equation of order δ1∈(3,4] with initial and boundary conditions, (Formula Presented), addressing the existence of a positive solution (EPS), where the fractional derivatives Dδ1, Dα1 are in the Riemann-Liouville sense of the order δ1, α1, respectively. The function (Formula Presented). To this aim, we establish an equivalent integral form of the problem with the help of a Green’s function. We also investigate the properties of the Green’s function in the paper which we utilize in our main result for the EPS of the problem. Results for the existence of solutions are obtained with the help of some classical results.
Citation Count: Khalil, Hammad...et al. (2019). "Some new operational matrices and its application to fractional order Poisson equations with integral type boundary constrains", Computers & Mathematics With Applications, Vol. 78, No. 6, pp. 1826-1837.
Some new operational matrices and its application to fractional order Poisson equations with integral type boundary constrains
(Pergamon-Elsevier Science LTD, 2019) Khalil, Hammad; Khan, Rahmat Ali; Baleanu, Dumitru; Rashidi, Mohammad Mehdi; 56389
Enormous application of fractional order partial differential equations (FPDEs) subjected to some constrains in the form of nonlocal boundary conditions motivated the interest of many scientists around the world. The prime objective of this article is to find approximate solution of a general FPDEs subject to nonlocal integral type boundary conditions on both ends of the domain. The proposed method is based on spectral method. We construct some new operational matrices which have the ability to handle integral type non-local boundary constrains. These operational matrices can be effectively applied to convert the FPDEs together with its integral types boundary conditions to easily solvable matrix equation. The accuracy and efficiency of proposed method are demonstrated by solving some bench mark problems. The proposed method has the ability to solve non-local FPDEs with high accuracy and low computational cost. Different aspects of presented approach are compared with two other recently developed methods, Haar wavelets collocation method and a family of collocation methods which are based on Radial base functions. It is observed that the proposed method is highly accurate, robust, efficient and stable as compared to these methods. (C) 2016 Elsevier Ltd. All rights reserved.
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: Samina; Shah...et al. (2019). "Study of implicit type coupled system of non-integer order differential equations with antiperiodic boundary conditions", Mathematical Methods in the Applied Sciences, Vol. 42, No. 6, pp. 2033-2042.
Study of implicit type coupled system of non-integer order differential equations with antiperiodic boundary conditions
(Wiley, 2019) Samina; Shah, Kamal; Khan, Rahmat Ali; Baleanu, Dumitru
In this paper, the first purpose is to study existence and uniqueness of solutions to a system of implicit fractional differential equations (IFDEs) equipped with antiperiodic boundary conditions (BCs). To obtain the mentioned results, we use Schauder's and Banach fixed point theorem. The second purpose is discussing the Ulam-Hyers (UH) and generalized Ulam-Hyers (GUH) stabilities for the respective solutions. An example is provided to illustrate the established results.