Matematik Bölümü Yayın Koleksiyonu
Permanent URI for this collectionhttps://hdl.handle.net/20.500.12416/413
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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: 17Citation - Scopus: 21Modeling the Transmission Dynamics of Delayed Pneumonia-Like Diseases With a Sensitivity of Parameters(Springer, 2021) Baleanu, Dumitru; Raza, Ali; Rafiq, Muhammad; Soori, Atif Hassan; Mohsin, Muhammad; Naveed, MuhammadPneumonia is a highly transmitted disease in children. According to the World Health Organization (WHO), the most affected regions include South Asia and sub-Saharan Africa. 15% deaths of children are due to pneumonia. In 2017, 0.88 million children were killed under the age of five years. An analysis of pneumonia disease is performed with the help of a delayed mathematical modelling technique. The epidemiological system contemplates subpopulations of susceptible, carriers, infected and recovered individuals, along with nonlinear interactions between the members of those subpopulations. The positivity and the boundedness of the ongoing problem for nonnegative initial data are thoroughly proved. The system possesses pneumonia-free and pneumonia existing equilibrium points, whose stability is studied rigorously. Moreover, the numerical simulations confirm the validity of these theoretical results.Article Citation - WoS: 17Citation - Scopus: 17Existence Results for a General Class of Sequential Hybrid Fractional Differential Equations(Springer, 2021) Gul, Shaista; Jarad, Fahd; Khan, Hasib; Khan, Rahmat AliIn 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.Article Citation - WoS: 22Citation - Scopus: 24Dynamics of Pattern Formation Process in Fractional-Order Super-Diffusive Processes: a Computational Approach(Springer, 2021) Karaagac, Berat; Baleanu, Dumitru; Owolabi, Kolade M.This paper explores the suitability of space fractional-order reaction-diffusion scenarios to model some emergent pattern formation in predator-prey models. Such fractional reaction-diffusion equations are obtained on the basis of a continuous-time random walk approach with spatial memory and local kinetic reaction. The classical space second-order derivative is changed by the fractional Laplacian case. We employ the Fourier spectral method to numerically approximate the fractional Laplacian and advance in time with the novel ETDRK4 method. In other to obtain guidelines on the correct choice of parameters when numerically simulating the full reaction-diffusion models, the local dynamics of the systems are considered. The biological wave scenarios of solutions are verified by presenting some numerical results in two dimensions to mimic some spatiotemporal dynamics such as spots, stripes and spiral patterns which has a lot of ecological implications.Article Citation - WoS: 32Citation - Scopus: 38An Efficient Numerical Scheme Based on Lucas Polynomials for the Study of Multidimensional Burgers-Type Equations(Springer, 2021) Haq, Sirajul; Nisar, Kottakkaran Sooppy; Baleanu, Dumitru; Ali, IhteramWe propose a polynomial-based numerical scheme for solving some important nonlinear partial differential equations (PDEs). In the proposed technique, the temporal part is discretized by finite difference method together with theta-weighted scheme. Then, for the approximation of spatial part of unknown function and its spatial derivatives, we use a mixed approach based on Lucas and Fibonacci polynomials. With the help of these approximations, we transform the nonlinear partial differential equation to a system of algebraic equations, which can be easily handled. We test the performance of the method on the generalized Burgers-Huxley and Burgers-Fisher equations, and one- and two-dimensional coupled Burgers equations. To compare the efficiency and accuracy of the proposed scheme, we computed L-infinity, L-2, and root mean square (RMS) error norms. Computations validate that the proposed method produces better results than other numerical methods. We also discussed and confirmed the stability of the technique.Article Citation - WoS: 66Citation - Scopus: 63Soliton Solutions and Stability Analysis for Some Conformable Nonlinear Partial Differential Equations in Mathematical Physics(Springer, 2018) Yusuf, Abdullahi; Aliyu, Aliyu Isa; Baleanu, Dumitru; Inc, MustafaThis research presents soliton solutions and stability analysis to some conformable nonlinear partial differential equations (CNPDEs). The CNPDEs equations in this paper are conformable Cahn-Allen and conformable Zoomeron equations. The powerful sine-Gordon method is used to carry out the soliton solutions for these equations. The aspects of stability analysis for the considered equations is investigated using the linear stability technique. The sine-Gordon method proves to be efficient and effective for the extraction of soliton solutions for different types of CNPDEs.Article Citation - WoS: 33Citation - Scopus: 37Extended Cubic B-Splines in the Numerical Solution of Time Fractional Telegraph Equation(Springer, 2019) Abbas, Muhammad; Ismail, Ahmad Izani; Ali, Norhashidah Hj M.; Baleanu, Dumitru; Akram, TayyabaA finite difference scheme based on extended cubic B-spline method for the solution of time fractional telegraph equation is presented and discussed. The Caputo fractional formula is used in the discretization of the time fractional derivative. A combination of the Caputo fractional derivative together with an extended cubic B-spline is utilized to obtain the computed solutions. The proposed scheme is shown to possess the unconditional stability property with second order convergence. Numerical results demonstrate the applicability, simplicity and the strength of the scheme in solving the time fractional telegraph equation with accuracies very close to the exact solutions.Article Citation - WoS: 25Citation - Scopus: 29Note on the Solution of Random Differential Equations Via Ψ-Hilfer Fractional Derivative(Springer, 2018) Shah, Kamal; Baleanu, Dumitru; Kanagarajan, K.; Harikrishnan, S.This manuscript is devoted to an investigation of the existence, uniqueness and stability of random differential equations with psi-Hilfer fractional derivative. The concerned investigation of existence and uniqueness is obtained via the Schauder fixed point theorem and Banach contraction principle, respectively. Furthermore, for the respective solutions, some results related to different kinds of Ulam type stability including Hyers-Ulam, and generalized Hyers-Ulam, Hyers-Ulam-Rassias are obtained.