Reproducing kernel Hilbert space method for the numerical solutions of fractional cancer tumor models

Attia, Nourhane; Akgül, Ali; Seba, Djamila; Nour, Abdelkader

doi:10.1002/mma.6940

Cited by 9 publications

(3 citation statements)

References 48 publications

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“…Contemporary data science, which revolves around diverse architectural frameworks of machine learning, signifies crucial potential across a wide range of applications. These applications span from various fields such as physics and engineering [1] to areas like genomic-assisted prediction [2] and predictive modeling for cancer tumors [3]. The central task of (automated) machine learning is to devise an optimal activation function or optimal kernel function, depending upon the learning architecture, that is best-suited for the specific scientific model.…”

Section: Introductionmentioning

confidence: 99%

Machine Learning Application of Generalized Gaussian Radial Basis Function and Its Reproducing Kernel Theory

Singh

2024

Mathematics

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Gaussian Radial Basis Function Kernels are the most-often-employed kernel function in artificial intelligence for providing the optimal results in contrast to their respective counterparts. However, our understanding surrounding the utilization of the Generalized Gaussian Radial Basis Function across different machine learning algorithms, such as kernel regression, support vector machines, and pattern recognition via neural networks is incomplete. The results delivered by the Generalized Gaussian Radial Basis Function Kernel in the previously mentioned applications remarkably outperforms those of the Gaussian Radial Basis Function Kernel, the Sigmoid function, and the ReLU function in terms of accuracy and misclassification. This article provides a concrete illustration of the utilization of the Generalized Gaussian Radial Basis Function Kernel as mentioned earlier. We also provide an explicit description of the reproducing kernel Hilbert space by embedding the Generalized Gaussian Radial Basis Function as an L2−measure, which is utilized in implementing the analysis support vector machine. Finally, we provide the conclusion that we draw from the empirical experiments considered in the manuscript along with the possible future directions in terms of spectral decomposition of the Generalized Gaussian Radial Basis Function.

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Section: Introductionmentioning

confidence: 99%

Machine Learning Application of Generalized Gaussian Radial Basis Function and Its Reproducing Kernel Theory

Singh

2024

Mathematics

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“…One can provide more realistic results and accurate predictions of virus dynamics using fractional derivatives. Besides mathematics, fractional derivatives have several applications in a variety of domains, including physics, finance, biology [18], medicine, ecology [19], and so on. There are several ways to define the fractional derivatives of a function, including the Riemann-Liouville definition, the Atangana-Baleanu derivative, and the Caputo fractional derivative.…”

Section: Introductionmentioning

confidence: 99%

Fractional‐order models of hepatitis B virus infection with recycling effects of capsids

Sutradhar,

Dalal

2023

Math Methods in App Sciences

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Hepatitis B virus (HBV) infection is a major public health concern throughout the world. It can be treated effectively through proper medication and vaccination, although it is very difficult to cure, especially in people who have had the chronic infection. So, it is necessary to pay proper attention for its eradication. In this study, two nonlinear fractional‐order HBV infection models with recycling effects of capsids are presented via the Caputo derivative. Each model contains four compartments, namely, susceptible hepatocytes, infected hepatocytes, HBV capsids, and viruses. In the first model, the order of fractional derivatives is equal for each compartment, whereas, in the second model, the order is incommensurate. The existence and uniqueness of solutions for both models are discussed separately. The parameters are estimated in order to validate the proposed model with experimental data obtained from a chimpanzee. Stability analyses are carried out for both models theoretically. The models are solved numerically using the predictor–corrector Adams–Bashforthm–Moulton method (for commensurate order) and implicit product integration of trapezoidal type method (for incommensurate order) with various choices of fractional orders and initial conditions. All the results are presented graphically. Interestingly, the results reveal the importance of fractional‐order derivatives in capturing the dynamics of HBV transmission in the host. It is also noticed that with the decrease in the order of fractional derivative, the peak level of infection decreases, but the disease takes a long time to be cured.

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“…It became a powerful tool in treating different types of FPDEs, such as the fractional Bloch-Torrey equations [40] and fractional differential equations, including the ABC derivative [41], to name a few. See also [42][43][44][45][46][47] for more research about this method. The RKHSM has many advantages, such as its simplicity and flexibility in treating many fractional differential systems and the fact that it is a mesh-free method.…”

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confidence: 99%

Analysis of the Fractional Differential Equations Using Two Different Methods

Partohaghighi

Akgül

et al. 2022

Symmetry

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Numerical methods play an important role in modern mathematical research, especially studying the symmetry analysis and obtaining the numerical solutions of fractional differential equation. In the current work, we use two numerical schemes to deal with fractional differential equations. In the first case, a combination of the group preserving scheme and fictitious time integration method (FTIM) is considered to solve the problem. Firstly, we applied the FTIM role, and then the GPS came to integrate the obtained new system using initial conditions. Figure and tables containing the solutions are provided. The tabulated numerical simulations are compared with the reproducing kernel Hilbert space method (RKHSM) as well as the exact solution. The methodology of RKHSM mainly relies on the right choice of the reproducing kernel functions. The results confirm that the FTIM finds the true solution. Additionally, these numerical results indicate the effectiveness of the proposed methods.

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Reproducing kernel Hilbert space method for the numerical solutions of fractional cancer tumor models

Cited by 9 publications

References 48 publications

Machine Learning Application of Generalized Gaussian Radial Basis Function and Its Reproducing Kernel Theory

Machine Learning Application of Generalized Gaussian Radial Basis Function and Its Reproducing Kernel Theory

Fractional‐order models of hepatitis B virus infection with recycling effects of capsids

Analysis of the Fractional Differential Equations Using Two Different Methods

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