Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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Article Two Numerical Solutions for Solving a Mathematical Model of the Avascular Tumor Growth(Dokuz Eylül Üniversitesi, 2021) Korkut, Sıla Övgü; İmamoğlu Karabaş, Neslişah; Başbınar, YaseminObjective: Cancer which is one of the most challenging health problems overall the world is composed of various processes: tumorigenesis, angiogenesis, and metastasis. Attempting to understand the truth behind this complicated disease is one of the common objectives of many experts and researchers from different fields. To provide deeper insights any prognostic and/or diagnostic scientific contribution to this topic is so crucial. In this study, the avascular tumor growth model which is the earliest stage of tumor growth is taken into account from a mathematical point of view. The main aim is to solve the mathematical model of avascular tumor growth numerically. Methods: This study has focused on the numerical solution of the continuum mathematical model of the avascular tumor growth described by Sharrett and Chaplin. Unlike the existing recent literature, the study has focused on the methods for the temporal domain. To obtain the numerical schemes the central difference method has been used in the spatial coordinates. This discretization technique has reduced the main partial differential equation into an ordinary differential equation which will be solved successively by two alternative techniques: the 4th order Runge-Kutta method (RK4) and the three-stage strongly-stability preserving Runge-Kutta method (SSP-RK3). Results: The model has been solved by the proposed methods. The numerical results are discussed in both mathematical and biological angles. The biological compatibility of the methods is depicted in various figures. Besides biological outputs, the accuracies of the methods have been listed from a mathematical point of view. Furthermore, the rate of convergence of the proposed methods has also been discussed computationally. Conclusion: All recorded results are evidence that the proposed schemes are applicable for solving such models. Moreover, all exhibited figures have proved the biological compatibility of the methods. It is observed that the quiescent cells which are one of the most mysterious cells in clinics tend to become proliferative for the selected parameters.Article A Reliable and Fast Mesh-Free Solver for the Telegraph Equation(Springer, 2022) İmamoğlu Karabaş, Neslişah; Korkut, Sıla Övgü; Gürarslan, Gürhan; Tanoğlu, GamzeIn the presented study, the hyperbolic telegraph equation is taken as the focus point. To solve such an equation, an accurate, reliable, and efficient method has been proposed. The developed method is mainly based on the combination of a kind of mesh-free method and an adaptive method. Multiquadric radial basis function mesh-free method is considered on spatial domain and the adaptive fifth-order Runge–Kutta method is used on time domain. The validity and the performance of the proposed method have been checked on several test problems. The approximate solutions are compared with the exact solution, it is shown that the proposed method has more preferable to the other methods in the literature.Article Citation - WoS: 2Citation - Scopus: 3A Reliable Explicit Method To Approximate the General Type of the Kdv–burgers’ Equation(Springer, 2022) Korkut, Sıla Övgü; İmamoğlu Karabaş, NeslişahThis study aims to propose a reliable, accurate, and efficient numerical approximation for a general compelling partial differential equation including nonlinearity (uδ∂u∂x), dissipation (∂2u∂x2), and dispersion (∂3u∂x3) which arises in many fields of engineering as well as applied sciences. The novel proposed method has been developed combining a kind of mesh-free method called the Taylor wavelet method with the Euler method. The convergence result of the method has been presented theoretically. Moreover, the validation and applicability of the method have been also confirmed computationally on benchmark problems such as KdV–Burgers’ equation and modified-KdV equation. The numerical results have been compared both to the exact solution and to those in the existing literature. All presented figures and tables guarantee that the proposed method is highly accurate, efficient, and compatible with the nature of the specified equation physically. Furthermore, the recorded errors are evidence that the proposed method is the best approximation compared to those in the existing methods.Article Citation - WoS: 10Citation - Scopus: 12An Efficient Approach for Solving Nonlinear Multidimensional Schrodinger Equations(Elsevier, 2021) İmamoğlu Karabaş, Neslişah; Korkut, Sıla Övgü; Tanoğlu, Gamze; Aziz, Imran; Siraj-ul-IslamAn efficient numerical method is proposed for the solution of the nonlinear cubic Schrodinger equation. The proposed method is based on the Frechet derivative and the meshless method with radial basis functions. An important characteristic of the method is that it can be extended from one-dimensional problems to multi-dimensional ones easily. By using the Frechet derivative and Newton-Raphson technique, the nonlinear equation is converted into a set of linear algebraic equations which are solved iteratively. Numerical examples reveal that the proposed method is efficient and reliable with respect to the accuracy and stability.Article Citation - Scopus: 2A Conserved Linearization Approach for Solving Nonlinear Oscillation Problems(Natural Sciences Publishing, 2018) Korkut, Sıla Övgü; Gücüyenen Kaymak, Nurcan; Tanoğlu, GamzeNonlinear oscillation problems are extensively used in engineering and applied sciences. Due to non-availability of the analytic solutions, numerical approaches have been used for these equations. In this study, a numerical method which is based on Newton-Raphson linearization and Fréchet derivative is suggested. The convergence analysis is also studied locally. The present method is tested on three examples: damped oscillator, Van-der Pol equation and Schrödinger equation. It is shown that the obtained solutions via the present method are more accurate than those of the well-known second order Runge-Kutta method. When examining the present method, preservation of characteristic properties of these equations is also considered. The obtained results show that the present method is applicable with respect to the efficiency and the physical compatibility.