Fractional Differential Equations

The computational work and storage of numerically solving the time fractional PDEs are generally huge for the traditional direct methods since they require total memory and work, where NT and NS represent the total number of time steps and grid points in space, respectively. To overcome this difficulty, we present an efficient algorithm for the evaluation of the Caputo fractional derivative of order α∈(0,1). The algorithm is based on an efficient sum-of-exponentials (SOE) approximation for the kernel t –1–α on the interval [Δt, T] with a uniform absolute error e. We give the theoretical analysis to show that the number of exponentials N exp needed is of order for T≫1 or for TH1 for fixed accuracy e. The resulting algorithm requires only storage and work when numerically solving the time fractional PDEs. Furthermore, we also give the stability and error analysis of the new scheme, and present several numerical examples to demonstrate the performance of our scheme.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/AF5FDC74FD7A010ED0ACD291BB1A92B1/S1815240617000147a.pdf/div-class-title-fast-evaluation-of-the-caputo-fractional-derivative-and-its-applications-to-fractional-diffusion-equations-div.pdf

Fast Evaluation of the Caputo Fractional Derivative and its Applications to Fractional Diffusion Equations

In this paper, a new alternating direction implicit Galerkin-Legendre spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation is developed. The temporal component is discretized by the Crank-Nicolson method. The detailed implementation of the method is presented. The stability and convergence analysis is strictly proven, which shows that the derived method is stable and convergent of order 2 in time. An optimal error estimate in space is also obtained by introducing a new orthogonal projector. The present method is extended to solve the fractional FitzHugh-Nagumo model. Numerical results are provided to verify the theoretical analysis.

/pdf/a-crank-nicolson-adi-spectral-method-for-a-two-dimensional-eg5dxd8kw4.pdf

A Crank-Nicolson adi spectral method for a two-dimensional Riesz space fractional nonlinear reaction-diffusion equation

Artificial intelligence (AI) has the potential to revolutionize the drug discovery process, offering improved efficiency, accuracy, and speed. However, the successful application of AI is dependent on the availability of high-quality data, the addressing of ethical concerns, and the recognition of the limitations of AI-based approaches. In this article, the benefits, challenges, and drawbacks of AI in this field are reviewed, and possible strategies and approaches for overcoming the present obstacles are proposed. The use of data augmentation, explainable AI, and the integration of AI with traditional experimental methods, as well as the potential advantages of AI in pharmaceutical research, are also discussed. Overall, this review highlights the potential of AI in drug discovery and provides insights into the challenges and opportunities for realizing its potential in this field. Note from the human authors: This article was created to test the ability of ChatGPT, a chatbot based on the GPT-3.5 language model, in terms of assisting human authors in writing review articles. The text generated by the AI following our instructions (see Supporting Information) was used as a starting point, and its ability to automatically generate content was evaluated. After conducting a thorough review, the human authors practically rewrote the manuscript, striving to maintain a balance between the original proposal and the scientific criteria. The advantages and limitations of using AI for this purpose are discussed in the last section.

/pdf/the-role-of-ai-in-drug-discovery-challenges-opportunities-39p1umjz.pdf

The Role of AI in Drug Discovery: Challenges, Opportunities, and Strategies

A fourth-order approximation of fractional derivatives with its applications

Fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second-order diffusion equations. Because of the nonlocal property of fractional differential operators, numerical methods for space-fractional diffusion equations generate complicated dense or full coefficient matrices. Consequently, these numerical methods were traditionally solved by Gaussian elimination, which requires computational work of $O(N^3)$ per time step and $O(N^2)$ of memory, where $N$ is the number of spatial grid points in the discretization. The significant computational work and memory requirement of the numerical methods impose a serious challenge for the numerical simulation of two- and especially three-dimensional space-fractional diffusion equations. We develop a fast and yet accurate solution method for the implicit finite difference discretization of space-fractional diffusion equations in two space dimensions by carefully analyzing the structure of the coefficient ma...

A Fast Finite Difference Method for Two-Dimensional Space-Fractional Diffusion Equations

Artificially intelligent computer systems are used extensively in medical sciences. Common applications include diagnosing patients, end-to-end drug discovery and development, improving communication between physician and patient, transcribing medical documents, such as prescriptions, and remotely treating patients. While computer systems often execute tasks more efficiently than humans, more recently, state-of-the-art computer algorithms have achieved accuracies which are at par with human experts in the field of medical sciences. Some speculate that it is only a matter of time before humans are completely replaced in certain roles within the medical sciences. The motivation of this article is to discuss the ways in which artificial intelligence is changing the landscape of medical science and to separate hype from reality.

Artificial Intelligence: How is It Changing Medical Sciences and Its Future?

Advances in Machine Learning and availability of state-of-the-art computational resources, along with digitized healthcare data, have set the stage for extensive application of artificial intelligence in the realm of diagnosis, prognosis, clinical decision support, personalized treatment options, drug development, and the field of biomedicine. Here, we discuss the application of Machine Learning algorithms in patient healthcare and dermatological domains along with the ethical complexities that are involved. In scientific studies, ethical challenges were initially not addressed proportionally (as assessed by keyword counts in PubMed) and just more recently (since 2016) this has started to improve. Few pioneering countries have created regulatory guidelines around how to respect matters of (1) privacy, (2) fairness, (3) accountability, (4) transparency and (5) conflict of interest when developing novel medical Machine Learning applications. While there is a strong promise of emerging medical applications to ultimately benefit both the patients and the medical practitioners, it is important to raise awareness on the five key ethical issues and incorporate them into medical practice in the near future.

The ethics of machine learning in medical sciences: Where do we stand today?

Stock trading has tremendous importance not just as a profession but also as an income source for individuals. Many investment account holders use the appreciation of their portfolio (as a combination of stocks or indexes) as income for their retirement years, mostly betting on stocks or indexes with low risk/low volatility. However, every stock-based investment portfolio has an inherent risk to lose money through negative progression and crash. This study presents a novel technique to predict such rare negative events in financial time series (e.g., a drop in the S&P 500 by a certain percent in a designated period of time). We use a time series of approximately seven years (2517 values) of the S&P 500 index stocks with publicly available features: the high, low and close price (HLC). We utilize a Siamese type neural network for pattern recognition in images followed by a bootstrapped image similarity distribution to predict rare events as they pertain to financial market analysis. Extending on literature about rare event classification and stochastic modeling in financial analytics, the proposed method uses a sliding window to store the input features as tabular data (HLC price), creates an image of the time series window, and then uses the feature vector of a pre-trained convolutional neural network (CNN) to leverage pre-event images and predict rare events. This research does not just indicate that our proposed method is capable of distinguishing event images from non-event images, but more importantly, the method is effective even when only limited and strongly imbalanced data is available.

https://www.mdpi.com/2227-9091/10/2/39/pdf?version=1645517642

A Novel Implementation of Siamese Type Neural Networks in Predicting Rare Fluctuations in Financial Time Series

This paper proposes an approach for the space-fractional diffusion equation in one dimension. Since fractional differential operators are non-local, two main difficulties arise after discretization and solving using Gaussian elimination: how to handle the memory requirement of O(N2) for storing the dense or even full matrices that arise from application of numerical methods and how to manage the significant computational work count of O(N3) per time step, where N is the number of spatial grid points. In this paper, a fast iterative finite difference method is developed, which has a memory requirement of O(N) and a computational cost of O(N logN) per iteration. Finally, some numerical results are shown to verify the accuracy and efficiency of the new method.

A Fast O(N log N) Finite Difference Method for the One-Dimensional Space-Fractional Diffusion Equation

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