Scopus İndeksli Yayınlar Koleksiyonu
Permanent URI for this collectionhttps://hdl.handle.net/20.500.12416/8651
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Article Citation - Scopus: 16Lucas Wavelet Scheme for Fractional Bagley–torvik Equations: Gauss–jacobi Approach(Springer, 2022) Koundal, R.; Kumar, R.; Srivastava, K.; Baleanu, D.A novel technique called as Lucas wavelet scheme (LWS) is prepared for the treatment of Bagley–Torvik equations (BTEs). Lucas wavelets for the approximation of unknown functions appearing in BTEs are introduced. Fractional derivatives are evaluated by employing Gauss–Jacobi quadrature formula. Further, well-known least square method (LSM) is adopted to compute the residual function, and the system of algebraic equation is obtained. Convergence criterion is derived and error bounds are obtained for the established technique. The scheme is investigated by choosing some reliable test problems through tables and figures, which ensures the convenience, validity and applicability of LWS. © 2021, The Author(s), under exclusive licence to Springer Nature India Private Limited.Article Citation - WoS: 2Citation - Scopus: 2Fractional Evolution Equation With Cauchy Data in L<sup>p</Sup> Spaces(Springer, 2022) Baleanu, Dumitru; Agarwal, Ravi P.; Le Dinh Long; Nguyen Duc Phuong; Long, Le Dinh; Phuong, Nguyen DucIn this paper, we consider the Cauchy problem for fractional evolution equations with the Caputo derivative. This problem is not well posed in the sense of Hadamard. There have been many results on this problem when data is noisy in L-2 and H-s,H- However, there have not been any papers dealing with this problem with observed data in L-p with p not equal 2. We study three cases of source functions: homogeneous case, inhomogeneous case, and nonlinear case. For all of them, we use a truncation method to give an approximate solution to the problem. Under different assumptions on the smoothness of the exact solution, we get error estimates between the regularized solution and the exact solution in L-p. To our knowledge, L-p evaluations for the inverse problem are very limited. This work generalizes some recent results on this problem.Article Citation - WoS: 13Citation - Scopus: 16A Robust Scheme for Caputo Variable-Order Time-Fractional Diffusion-Type Equations(Springer, 2023) Hosseini, Kamyar; Baleanu, Dumitru; Salahshour, Soheil; Hincal, Evren; Sadri, KhadijehThe focus of this work is to construct a pseudo-operational Jacobi collocation scheme for numerically solving the Caputo variable-order time-fractional diffusion-type equations with applications in applied sciences. Modeling scientific phenomena in the context of fluid flow problems, curing reactions of thermosetting systems, solid oxide fuel cells, and solvent diffusion into heavy oils led to the appearance of these equations. For this reason, the numerical solution of these equations has attracted a lot of attention. More precisely, using pseudo-operational matrices and appropriate approximations based on bivariate Jacobi polynomials, the approximate solutions of the variable-order time-fractional diffusion-type equations in the Caputo sense with high accuracy are formally retrieved. Based on orthogonal bivariate Jacobi polynomials and their operational matrices, a sparse algebraic system is generated which makes implementing the proposed approach easy. An error bound is computed for the residual function by proving some theorems. To illustrate the accuracy and efficiency of the scheme, several illustrative examples are considered. The results demonstrate the efficiency of the present method compared to those achieved by the Legendre and Lucas multi-wavelet methods and the Crank-Nicolson compact method.Article Citation - WoS: 68Citation - Scopus: 74On Continuity of the Fractional Derivative of the Time-Fractional Semilinear Pseudo-Parabolic Systems(Springer, 2021) Ho Duy Binh; Nguyen Hoang Luc; Nguyen Huu Can; Karapinar, Erdal; Binh, Ho Duy; Luc, Nguyen Hoang; Can, Nguyen HuuIn this work, we study an initial value problem for a system of nonlinear parabolic pseudo equations with Caputo fractional derivative. Here, we discuss the continuity which is related to a fractional order derivative. To overcome some of the difficulties of this problem, we need to evaluate the relevant quantities of the Mittag-Leffler function by constants independent of the derivative order. Moreover, we present an example to illustrate the theory.Article Citation - WoS: 25Citation - Scopus: 22Numerical Approximation of Inhomogeneous Time Fractional Burgers-Huxley Equation With B-Spline Functions and Caputo Derivative(Springer, 2022) Kamran, Mohsin; Asghar, Noreen; Baleanu, Dumitru; Majeed, AbdulA prototype model used to explain the relationship between mechanisms of reaction, convection effects, and transportation of diffusion is the generalized Burgers-Huxley equation. This study presents numerical solution of non-linear inhomogeneous time fractional Burgers-Huxley equation using cubic B-spline collocation method. For this purpose, Caputo derivative is used for the temporal derivative which is discretized by L1 formula and spatial derivative is interpolated with the help of B-spline basis functions, so the dependent variable is continuous throughout the solution range. The validity of the proposed scheme is examined by solving four test problems with different initial-boundary conditions. The algorithm for the execution of scheme is also presented. The effect of non-integer parameter alpha and time on dependent variable is studied. Moreover, convergence and stability of the proposed scheme is analyzed, and proved that scheme is unconditionally stable. The accuracy is checked by error norms. Based on obtained results we can say that the proposed scheme is a good addition to the existing schemes for such real-life problems.Article Citation - WoS: 16Citation - Scopus: 19Finite-Time Stability of Linear Stochastic Fractional-Order Systems With Time Delay(Springer, 2021) Ben Makhlouf, Abdellatif; Baleanu, Dumitru; Rhaima, Mohamed; Mchiri, LassaadThis paper focuses on the finite-time stability of linear stochastic fractional-order systems with time delay for alpha is an element of (1/2, 1). Under the generalized Gronwall inequality and stochastic analysis techniques, the finite-time stability of the solution for linear stochastic fractional-order systems with time delay is investigated. We give two illustrative examples to show the interest of the main results.Article Citation - WoS: 11Citation - Scopus: 17Bivariate Chebyshev Polynomials of the Fifth Kind for Variable-Order Time-Fractional Partial Integro-Differential Equations With Weakly Singular Kernel(Springer, 2021) Hosseini, Kamyar; Baleanu, Dumitru; Ahmadian, Ali; Salahshour, Soheil; Sadri, KhadijehThe shifted Chebyshev polynomials of the fifth kind (SCPFK) and the collocation method are employed to achieve approximate solutions of a category of the functional equations, namely variable-order time-fractional weakly singular partial integro-differential equations (VTFWSPIDEs). A pseudo-operational matrix (POM) approach is developed for the numerical solution of the problem under study. The suggested method changes solving the VTFWSPIDE into the solution of a system of linear algebraic equations. Error bounds of the approximate solutions are obtained, and the application of the proposed scheme is examined on five problems. The results confirm the applicability and high accuracy of the method for the numerical solution of fractional singular partial integro-differential equations.Article Citation - WoS: 6Citation - Scopus: 8Analytic and Numerical Solutions of Discrete Bagley-Torvik Equation(Springer, 2021) Khashan, M. Motawi; Xavier, Gnanaprakasam Britto Antony; Jarad, Fahd; Meganathan, Murugesan; Abdeljawad, Thabet; Britto Antony Xavier, Gnanaprakasam; Motawi Khashan, M.In this research article, a discrete version of the fractional Bagley-Torvik equation is proposed: del(2)(h)u(t) + A(C)del(nu)(h) u(t) + Bu(t) = f (t), t > 0, (1) where 0 < nu < 1 or 1 < nu < 2, subject to u(0) = a and del(h)u(0) = b, with a and b being real numbers. The solutions are obtained by employing the nabla discrete Laplace transform. These solutions are expressed in terms of Mittag-Leffler functions with three parameters. These solutions are handled numerically for some examples with specific values of some parameters.Article Citation - WoS: 37Citation - Scopus: 45The Operational Matrix Formulation of the Jacobi Tau Approximation for Space Fractional Diffusion Equation(Springer, 2014) Bhrawy, Ali H.; Baleanu, Dumitru; Ezz-Eldien, Samer S.; Doha, Eid H.In this article, an accurate and efficient numerical method is presented for solving the space-fractional order diffusion equation (SFDE). Jacobi polynomials are used to approximate the solution of the equation as a base of the tau spectral method which is based on the Jacobi operational matrices of fractional derivative and integration. The main advantage of this method is based upon reducing the nonlinear partial differential equation into a system of algebraic equations in the expansion coefficient of the solution. In order to test the accuracy and efficiency of our method, the solutions of the examples presented are introduced in the form of tables to make a comparison with those obtained by other methods and with the exact solutions easy.Article Citation - WoS: 17Citation - Scopus: 25Ulam Stability Results To a Class of Nonlinear Implicit Boundary Value Problems of Impulsive Fractional Differential Equations(Springer, 2019) Shah, K.; Baleanu, D.; Ali, A.In this paper, we derive some sufficient conditions which ensure the existence and uniqueness of a solution for a class of nonlinear three point boundary value problems of fractional order implicit differential equations (FOIDEs) with some boundary and impulsive conditions. Also we investigate various types of Hyers-Ulam stability (HUS) for our concerned problem. Using classical fixed point theory and nonlinear functional analysis, we obtain the required conditions. In the last section we give an example to show the applicability of our obtained results.