Imagine you have a huge library of books, and you need to find a specific piece of information from each book. Now, instead of manually searching through each book, what if you had a super smart assistant who could do the job for you? Well, that’s kind of what scientists have done with integral equations using a neural network algorithm based on the sine-cosine basis function and extreme learning machine (ELM). They created a new method to solve Volterra and Fredholm integral equations by designing a special neural network model. This model has layers that work together like an assembly line, with each layer doing its own unique job. By utilizing the sine-cosine basis function and the ELM algorithm, they were able to simplify the model and make it faster and more efficient. The best part? This method can not only solve different types of integral equations accurately but also provide solutions in a continuous and differentiable form. So, if you’re interested in learning more about this fascinating research and how it can revolutionize problem-solving in mathematics, be sure to check out the full article!

In this study, we investigate a new neural network method to solve Volterra and Fredholm integral equations based on the sine-cosine basis function and extreme learning machine (ELM) algorithm. Considering the ELM algorithm, sine-cosine basis functions, and several classes of integral equations, the improved model is designed. The novel neural network model consists of an input layer, a hidden layer, and an output layer, in which the hidden layer is eliminated by utilizing the sine-cosine basis function. Meanwhile, by using the characteristics of the ELM algorithm that the hidden layer biases and the input weights of the input and hidden layers are fully automatically implemented without iterative tuning, we can greatly reduce the model complexity and improve the calculation speed. Furthermore, the problem of finding network parameters is converted into solving a set of linear equations. One advantage of this method is that not only we can obtain good numerical solutions for the first- and second-kind Volterra integral equations but also we can obtain acceptable solutions for the first- and second-kind Fredholm integral equations and Volterra–Fredholm integral equations. Another advantage is that the improved algorithm provides the approximate solution of several kinds of linear integral equations in closed form (i.e., continuous and differentiable). Thus, we can obtain the solution at any point. Several numerical experiments are performed to solve various types of integral equations for illustrating the reliability and efficiency of the proposed method. Experimental results verify that the proposed method can achieve a very high accuracy and strong generalization ability.

Read Full Article (External Site)