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Analysis and numerical computations of the fractional regularized long-wave equation with damping term
(Wiley, 2021) Yavuz, Mehmet; Sulaiman, Tukur Abdulkadir; Usta, Fuat; Bulut, Hasan
This study explores the fractional damped generalized regularized long-wave equation in the sense of Caputo, Atangana-Baleanu, and Caputo-Fabrizio fractional derivatives. With the aid of fixed-point theorem in the Atangana-Baleanu fractional derivative with Mittag-Leffler-type kernel, we show the existence and uniqueness of the solution to the damped generalized regularized long-wave equation. The modified Laplace decomposition method (MLDM) defined in the sense of Caputo, Atangana-Baleanu, and Caputo-Fabrizio (in the Riemann sense) operators is used in securing the approximate-analytical solutions of the nonlinear model. The numerical simulations of the obtained solutions are performed with different suitable values of rho, which is the order of fractional parameter. We have seen the effect of the various parameters and variables on the displacement in figures.
The analytical solution of Van der Pol and Lienard differential equations within conformable fractional operator by retarded integral inequalities
(Sciendo, 2019) Usta, Fuat; Sarıkaya, Mehmet Zeki
In this study we introduced and tested retarded conformable fractional integral inequalities utilizing non-integer order derivatives and integrals. In line with this purpose, we used the Katugampola type conformable fractional calculus which has several practical properties. These inequalities generalize some famous integral inequalities which provide explicit bounds on unknown functions. The results provided here had been implemented to the global existence of solutions to the conformable fractional differential equations with time delay.
Approximating the Finite Hilbert Transform for Absolutely Continuous Mappings and Applications in Numerical Integration
(Springer Basel Ag, 2018) Usta, Fuat
The finite Hilbert transform is a practical instrument in the field of signal processing, time series analysis, radar systems, and other fields of the engineering sciences. In this study, some explicit bounds for the finite Hilbert transform are given utilizing the fundamental integral identity for absolutely continuous mappings.
Approximating the Finite Mellin and Sumudu Transforms Utilizing Wavelet Transform
(Univ Nis, Fac Sci Math, 2020) Usta, Fuat; Budak, Huseyin; Sarikaya, Mehmet Zeki
In this study, some approximates for the finite Wavelet transform of different classes of absolutely continues mappings are presented using Wavelet transform of unit function. Then, with the help of these approximates, some other approximates for the finite Mellin and Sumudu transforms are given.
Approximation of functions by a new class of Gamma type operators; theory and applications
(Ovidius Univ Press, 2024) Ozcelik, Reyhan; Kara, Emrah Evren; Usta, Fuat
The study of the linear methods of approximation, which are given by sequences of positive and linear operators, studied extremely, in relation to different subjects of analysis, such as numerical analysis. The principal objective of this manuscript is to develop a new and more comprehensive version of Gamma type operators and presented their approximation features. For this purpose, we benefit from two sequences of functions, which are alpha(n)(x) and beta(n)(x), and from the function tau(x). To indicate how the function tau play a significant role in the construction of the operator, we reconstruct the mentioned operators which preserve exactly two test functions from the set {1, tau, tau(2)}. Then we established Voronovskaya type theorem and order of approximation properties of the newly defined operators utilizing weighted modulus of continuity to show that their approximation properties. At the end of this note, we present a series of numerical results to show that the new operators are an approximation technique.
Approximation of functions by a new construction ofBernstein-Chlodowskyoperators: Theory and applications
(Wiley, 2021) Usta, Fuat
The main motivation of this paper is to provide a generalization of Bernstein-Chlodowsky type operators which depend on function tau by means of two sequences of functions. The newly defined operators fix the test function set{1, tau, tau(2)}. Then we present the approximation properties of newly defined operators, such as weighted approximation, degree of approximation and Voronovskaya type theorems. Finally, we present a series of numerical examples demonstrating the effectiveness of this newly defined Bernstein-Chlodowsky operators for computing function approximation.
Approximation of functions by a new type of Gamma operators
(Wiley, 2020) Betus, Omur; Usta, Fuat
In this article, a new type of Gamma operators which preserves 1 and x(2) has been presented and tested. In order to show the approximation properties of the newly defined operator, Voronovksya type theorems, weighted approximation, rate of convergence, and pointwise estimates have been presented. Additionally, numerical examples implemented by MATLAB have been provided to show its approximation properties by numerically.
Approximation of functions with linear positive operators which fix {1, phi} and {1, phi(2)}
(Ovidius Univ Press, 2020) Usta, Fuat
In this manuscript, linear and positive operators described on bounded and unbounded intervals that fix the function sets {1, phi} and {1, phi(2)} such that phi is an element of C[0, 1] are presented. Then we present different types of operators by choosing different functions and values. Finally, Voronovskaya type theorems are given for this newly defined sequences of linear and positive operators.
Approximation properties of a new family of Gamma operators and their applications
(Springer, 2021) Özçelik, Reyhan; Kara, Emrah Evren; Usta, Fuat; Ansari, Khursheed J.
The present paper introduces a new modification of Gamma operators that protects polynomials in the sense of the Bohman-Korovkin theorem. In order to examine their approximation behaviours, the approximation properties of the newly introduced operators such as Voronovskaya-type theorems, rate of convergence, weighted approximation, and pointwise estimates are presented. Finally, we present some numerical examples to verify that the newly constructed operators are an approximation procedure.
Approximation properties of Bernstein-Stancu operators preserving e?2x
(University of Nis, 2023) Usta, Fuat; Mursaleen, M.; Çakır, İ.
Bernstein-Stancu operators are one of the most powerful tool that can be used in approximation theory. In this manuscript, we propose a new construction of Bernstein-Stancu operators which preserve the constant and e?2x, x > 0. In this direction, the approximation properties of this newly defined operators have been examined in the sense of different function spaces. In addition to these, we present the Voronovskaya type theorem for this operators. At the end, we provide two computational examples to demonstrate that the new operator is an approximation procedure. © 2023, University of Nis. All rights reserved.
Approximation theorems for the new construction of Balazs operators and its applications
(Walter De Gruyter Gmbh, 2022) Usta, Fuat
Balazs operators are an influential tool that can be used to approximate a function on the unbounded interval [0, infinity). In this study, a new construction of Balazs operators which depending upon on a function rho(x) has been introduced. The function rho(x) plays a significant role in this construction due to the fact that the new operator preserves definitely two test functions from the set of {1, rho(x), rho(2)(x)}. In this direction, the approximation properties of this newly defined operators are established, such as degree of approximation and Voronovskaya type theorems. Finally, we presented a set of computational examples in order to validate the new construction of operator is an approximation procedure.
Bernstein approximation technique for numerical solution of Volterra integral equations of the third kind
(Springer Heidelberg, 2021) Usta, Fuat
We propose a numerical scheme based upon the Bernstein approximation method for computational solution of a new class of Volterra integral equations of the third kind (3rdVIEs). Construction of the technique and its practicality for proposed equations have been introduced. Furthermore we have examined the numerability and convergence analysis of the proposed scheme. Finally, we demonstrate a series of numerical examples demonstrating the effectiveness of this new technique for solving 3rdVIEs.
Compact Operators on Generalized Fibonacci Spaces
(Amer Inst Physics, 2019) İlkhan, Merve; Usta, Fuat; Kara, Emrah Evren
The main purpose of this paper is to characterize compactness of certain matrix operators on the generalized Fibonacci space by using the Hausdorff measure of non-compactness.
Compact operators on the Jordan totient sequence spaces
(Wiley, 2021) Ilkhan, Merve; Kara, Evren Emrah; Usta, Fuat
The necessary and sufficient conditions for compactness of a matrix operator between Banach spaces is obtained by utilizing the concept of the Hausdorff measure of noncompactness. This is one of the most interesting application in the theory of sequence spaces. In this paper, the compact operators are characterized on Jordan totient sequence spaces by using the concept of the Hausdorff measure of noncompactness.
Computational Solution of Katugampola Conformable Fractional Differential Equations via RBF Collocation Method
(Amer Inst Physics, 2017) Usta, Fuat
In conjunction with the development of fractional calculus, conformable derivatives and integrals has been widely used a number of scientific areas. In this talk, we provide a numerical scheme to solve Katugampola conformable fractional differential equations via radial basis function (RBF) collocation technique. In order to confirm our numerical scheme, we present some numerical experiments results.
A conformable calculus of radial basis functions and its applications
(Balikesir University, 2018) Usta, Fuat
In this paper we introduced the conformable derivatives and integrals of radial basis functions (RBF) to solve conformable fractional differential equations via RBF collocation method. For that, firstly, we found the conformable derivatives and integrals of power, Gaussian and multiquadric basis functions utilizing the rule of conformable fractional calculus. Then by using these derivatives and integrals we provide a numerical scheme to solve conformable fractional differential equations. Finally we presents some numerical results to confirmed our method. © 2018 Balikesir University. All rights reserved.
Editorial for special issue IECMSA 2020-International Eurasian Conference on Mathematical Sciences and Applications
(Wiley, 2022) Tosun, Murat; Kara, Emrah Evren; Usta, Fuat
[Bastract Not Available]
Effect of the Awareness Parameter on a Fractional-Order Tuberculosis Model
(American Institute of Physics Inc., 2022) Yavuz, M.; Akman, M.; Usta, Fuat; Özdemir, N.
Tuberculosis (TB) is an infectious disease with a high death rate compared to many infectious diseases. Therefore, many prominent studies have been done on the mathematical modeling and analysis of TB. In this study, an illustrative mathematical model is developed by considering the awareness parameter. In this context, two different treatment strategies that is applied as protective treatment and main treatment that is considered on the infected individuals are taken into account. In the provided model, a six-dimensional compartment system of fractional-order is constructed that includes the susceptible, latent, infected, and recovered population, as well as including the mentioned two treatment strategies. Also positivity and biologically feasible region of the model are provided. In the numerical simulations, Adams-Bashforth which is a well-known numerical scheme is applied to obtain the results. © 2022 American Institute of Physics Inc.. All rights reserved.
Explicit bounds on certain integral inequalities via conformable fractional calculus
(Taylor & Francis As, 2017) Usta, Fuat; Sarıkaya, Mehmet Zeki
In this paper, we present some explicit upper bounds for integral inequalities with the help of Katugampola-type conformable fractional calculus. The results have been obtained to cover the previous published studies for Gronwall-Bellman and Bihari like integral inequalities.
FRACTIONAL TYPE POISSON EQUATIONS BY RADIAL BASIS FUNCTIONS KANSA APPROACH
(Univ Prishtines, 2016) Usta, Fuat
In this paper we propose a numerical scheme for the solution of fractional order of Poisson equation in R-2. The new scheme uses the Radial basis functions (RBFs) method to benefit the desired properties of mesh free techniques such as no need to generate any mesh and easily applied to high dimensions. In the numerical solution approach the Kansa's collocation method is used to discrete fractional derivative terms with the multiquadric basis function. The numerical experiments two dimensional cases are presented and discussed, which conform well with the corresponding exact solutions.

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