Matematik Bölümü Yayın Koleksiyonu
Permanent URI for this collectionhttps://hdl.handle.net/20.500.12416/413
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Article Some more bounded and singular pulses of a generalized scale-invariant analogue of the Korteweg–de Vries equation(2023) Saifullah, Sayed; Alqarni, M.M.; Ahmad, Shabir; Baleanu, Dumitru; Khan, Meraj Ali; Mahmoud, Emad E.We investigate a generalized scale-invariant analogue of the Korteweg–de Vries (KdV) equation, establishing a connection with the recently discovered short-wave intermediate dispersive variable (SIdV) equation. To conduct a comprehensive analysis, we employ the Generalized Kudryashov Technique (KT), Modified KT, and the sine–cosine method. Through the application of these advanced methods, a diverse range of traveling wave solutions is derived, encompassing both bounded and singular types. Among these solutions are dark and bell-shaped waves, as well as periodic waves. Significantly, our investigation reveals novel solutions that have not been previously documented in existing literature. These findings present novel contributions to the field and offer potential applications in various physical phenomena, enhancing our understanding of nonlinear wave equations.Article Solitary wave solutions to Gardner equation using improved tan(Ω(Υ)/2-expansion method(2023) Akram, Ghazala; Sadaf, Maasoomah; Dawood, Mirfa; Abbas, Muhammad; Baleanu, DumitruIn this study, the improved tan(Ω(Υ)/2-expansion method is used to construct a variety of precise soliton and other solitary wave solutions of the Gardner equation. Gardner equation is extensively utilized in plasma physics, quantum field theory, solid-state physics and fluid dynamics. It is the simplest model for the description of water waves with dual power law nonlinearity. Hyperbolic, exponential, rational and trigonometric traveling wave solutions are obtained. The retrieved solutions include kink solitons, bright solitons, dark-bright solitons and periodic wave solutions. The efficacy of this method is determined by the comparison of the newly obtained results with already reported results.Article Citation - WoS: 3Citation - Scopus: 3Simulating systems of Ito? SDEs with split-step (?, ?)-Milstein scheme(Amer Inst Mathematical Sciences-AIMS, 2022) Ranjbar, Hassan; Torkzadeh, Leila; Baleanu, Dumitru; Nouri, KazemIn the present study, we provide a new approximation scheme for solving stochastic differential equations based on the explicit Milstein scheme. Under sufficient conditions, we prove that the split-step (alpha, beta)-Milstein scheme strongly convergence to the exact solution with order 1.0 in mean-square sense. The mean-square stability of our scheme for a linear stochastic differential equation with single and multiplicative commutative noise terms is studied. Stability analysis shows that the mean-square stability of our proposed scheme contains the mean-square stability region of the linear scalar test equation for suitable values of parameters alpha, beta. Finally, numerical examples illustrate the effectiveness of the theoretical results.Conference Object On Weighted Fractional Operators with Applications to Mathematical Models Arising in Physics(2023) Samraiz, Muhammad; Umer, Muhammad; Naheed, Saima; Baleanu, DumitruIn recent study, we develop the weighted generalized Hilfer-Prabhakar fractional derivative operator and explore its key properties. It unifies many existing fractional derivatives like Hilfer-Prabhakar and Riemann-Liouville. The weighted Laplace transform of the newly defined derivative is obtained. By involving the new fractional derivative, we modeled the free-electron laser equation and kinetic equation and then found the solutions of these fractional equations by applying the weighted Laplace transform.Article Citation - WoS: 13Citation - Scopus: 17Nonautonomous lump-periodic and analytical solutions tothe (3+1)-dimensional generalized Kadomtsev-Petviashviliequation(Springer, 2023) Alquran, Marwan; Sulaiman, Tukur Abdulkadir; Yusuf, Abdullahi; Alshomrani, Ali S.; Baleanu, DumitruThis work establishes the lump periodic and exact traveling wave solutions for the (3 + 1)-dimensional generalized Kadomtsev-Petviashvili equation. We use the Hirota bilinear method, as well as the robust integration techniques tanh-coth expansion and rational sine-cosine, to provide such innovative solutions. In order to explain specific physical difficulties, innovative lump periodic and analytical solutions have been investigated. These discoveries have been proven to be useful in the transmission of long-wave and high-power communications networks. It is important to highlight that the results given in thiswork depict new features and reflect previously unknown physical dynamics for the governing model.Article Impulsive Fractional Differential Equations with Retardation and Anticipation(2023) Benchohra, Mouffak; Karapınar, Erdal; Lazreg, Jamal Eddine; Salim, AbdelkrimThis chapter deals with the existence and uniqueness results for a class of impulsive initial and boundary value problems for implicit nonlinear fractional differential equations and k-Generalized ψ -Hilfer fractional derivative involving both retarded and advanced arguments. Our results are based on some necessary fixed point theorems. Suitable illustrative examples are provided for each section.Conference Object Citation - Scopus: 1Fractional Order Computing and Modeling with Portending Complex Fit Real-World Data(Springer International Publishing AG, 2023) Karaca, Yeliz; Rahman, Mati ur; Baleanu, DumitruFractional computing models identify the states of different systems with a focus on formulating fractional order compartment models through the consideration of differential equations based on the underlying stochastic processes. Thus, a systematic approach to address and ensure predictive accuracy allows that the model remains physically reasonable at all times, providing a convenient interpretation and feasible design regarding all the parameters of the model. Towards these manifolding processes, this study aims to introduce new concepts of fractional calculus that manifest crossover effects in dynamical models. Piecewise global fractional derivatives in sense of Caputo and Atangana-Baleanu-Caputo (ABC) have been utilized, and they are applied to formulate the Zika Virus (ZV) disease model. To have a predictive analysis of the behavior of the model, the domain is subsequently split into two subintervals and the piecewise behavior is investigated. Afterwards, the fixed point theory of Schauder and Banach is benefited from to prove the existence and uniqueness of at least one solution in both senses for the considered problem. As for the numerical simulations as per the data, Newton interpolation formula has been modified and extended for the considered nonlinear system. Finally, graphical presentations and illustrative examples based on the data for various compartments of the systems have been presented with respect to the applicable real-world data for different fractional orders. Based on the impact of fractional order reducing the abrupt changes, the results obtained from the study demonstrate and also validate that increasing the fractional order brings about a greater crossover effect, which is obvious from the observed data, which is critical for the effective management and control of abrupt changes like infectious diseases, viruses, among many more unexpected phenomena in chaotic, uncertain and transient circumstances.Book Part Citation - Scopus: 3Fractional Gegenbauer Kernel Functions: Theory and Application(Springer, 2023) Nedaei Janbesaraei, Sherwin; Azmoon, Amirreza; Baleanu, DumitruBecause of the usage of many functions as a kernel, the support vector machine method has demonstrated remarkable versatility in tackling numerous machine learning issues. Gegenbauer polynomials, like the Chebyshev and Legender polynomials which are introduced in previous chapters, are among the most commonly utilized orthogonal polynomials that have produced outstanding results in the support vector machine method. In this chapter, some essential properties of Gegenbauer and fractional Gegenbauer functions are presented and reviewed, followed by the kernels of these functions, which are introduced and validated. Finally, the performance of these functions in addressing two issues (two example datasets) is evaluated.Article Citation - WoS: 11Citation - Scopus: 18Analysis of the family of integral equation involving incomplete types of I and Ī-functions(Taylor & Francis Ltd, 2023) Bhatter, Sanjay; Jangid, Kamlesh; Kumawat, Shyamsunder; Baleanu, Dumitru; Suthar, D.L.; Purohit, Sunil DuttThe present article introduces and studies the Fredholm-type integral equation with an incomplete I-function (IIF) and an incomplete (Formula presented.) -function (I (Formula presented.) F) in its kernel. First, using fractional calculus and the Mellin transform principle, we solve an integral problem involving IIF. The idea of the Mellin transform and fractional calculus is then used to analyse an integral equation using the incomplete (Formula presented.) -function. This is followed by the discovery and investigation of several important exceptional cases. This article's general discoveries may yield new integral equations and solutions. The desired outcomes seem to be very helpful in resolving many real-world problems whose solutions represent different physical phenomena. And also, findings help solve introdifferential, fractional differential, and extended integral equation problems.Article An e ffective method for solving nonlinear integral equations involving the Riemann-Liouville fractional operator(2023) Paul, Supriya Kumar; Mishra, Lakshmi Narayan; Mishra, Vishnu Narayan; Baleanu, DumitruIn this paper, under some conditions in the Banach space C([0; beta];R), we establish the existence and uniqueness of the solution for the nonlinear integral equations involving the Riemann-Liouville fractional operator (RLFO). To establish the requirements for the existence and uniqueness of solutions, we apply the Leray-Schauder alternative and Banach's fixed point theorem. We analyze Hyers-Ulam-Rassias (H-U-R) and Hyers-Ulam (H-U) stability for the considered integral equations involving the RLFO in the space C([0; beta];R). Also, we propose an e ffective and e fficient computational method based on Laguerre polynomials to get the approximate numerical solutions of integral equations involving the RLFO. Five examples are given to interpret the method.