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In this paper, we present new identities involving the biperiodic Fibonacci and Lucas sequences. Then we solve completely some quadratic Diophantine equations involving the bi-periodic Fibonacci and Lucas sequences. [ABSTRACT FROM AUTHOR]

A number N is a triangular number if it can be written as N=0+1+....+n for some natural number n. We study the problem of finding all nonnegative integer solutions of the Diophantine equation (0+1+....+x)+(0+1+....+y)=(0+1+...+z).Using this equation, some new and curious increasing integer sequences are built. [ABSTRACT FROM AUTHOR]

A class of nonlinear difference equations is considered. We show how their polynomial solutions can be computed in a systematic manner. [ABSTRACT FROM AUTHOR]

In this article, we study differential equations of the form y' = ∑Ai(x)yi / ∑Bi(x)yi which can be elliptic, hyperbolic, parabolic, Riccati, or quasi-linear. We show how rational solutions can be computed in a systematic manner. Such results are most likely to find applications in the theory of limit cycles as indicated by Giné et al. [ABSTRACT FROM AUTHOR]

In this paper, a reliable soft computing framework is presented for the approximate solution of initial value problem (IVP) of first Painlevé equation using three unsupervised neural network models optimized with sequential quadratic programming (SQP). These mathematical models are constructed in the form of feed-forward architecture including log-sigmoid, radial base and tan-sigmoid activation functions in the hidden layers. The optimization of designed parameters for each model is performed with SQP, an efficient constraint optimization problem-solving algorithm. The designed methodology is tested on the IVP, and comparative study is carried out with standard solution based on numerical and analytical solvers. The accuracy, convergence and effectiveness of the schemes are validated on the given benchmark problem by large number of simulations and their comprehensive analysis. [ABSTRACT FROM AUTHOR]