Abstract:This research presents the solution of the generalized version of Abel’s integral equation, which was computed considering the first and second kinds. First, Abel’s integral equation and its generalization were described using fractional calculus, and the properties of Orthogonal polynomials were also described. We then developed a technique of solution for the generalized Abel’s integral equation using infinite series of orthogonal polynomials and utilized the numerical method to approximate the generalized Abel’s integral equation of the first and second kind, respectively. The Riemann-Liouville fractional operator was used in these examples. Our technique was implemented in MAPLE 17 through some illustrative examples. Absolute errors were estimated. In addition, the occurred errors between using orthogonal polynomials for solving Abel’s integral equations of order \(0\ <\ \alpha \ <\ 1\) and the exact solutions show that the orthogonal polynomials used were highly effective, reliable and can be used independently in situations where the exact solution is unknown which the numerical experiments confirmed.

Abstract:Stochastic differential equations (SDEs) are a powerful tool for modeling certain random trajectories of diffusion phenomena in the physical, ecological, economic, and management sciences. However, except in some cases, it is generally impossible to find an explicit solution to these equations. In this case, the numerical approach is the only favorable possibility to find an approximative solution. In this paper, we present the mean and mean-square stability of the Non-standard Euler-Maruyama numerical scheme using the Vasicek and geometric Brownian motion models.

Abstract:Using opoola differential operator, we defined a subclass \(S^{n}_{p}(\lambda,\alpha,\gamma,\delta)\) of the class of multivalent or p-valent functions. Several properties of the class were studied, such as coefficient inequalities, hadamard product, radii of close-to-convex, star-likeness, convexity, extreme points, the integral mean inequalities for the fractional derivatives, and further growth and distortion theorem are given using fractional calculus techniques.

Abstract:In this paper, we present results of \(\omega\)-order preserving partial contraction mapping generating a nonlinear Schr\”odinger equation. We used the theory of semigroup to generate a nonlinear Schr\(\ddot{o}\)dinger equation by considering a simple application of Lipschitz perturbation of linear evolution equations. We considered the space \(L^2(\mathbb{R}^2)\) and of linear operator \(A_0$ by $D(A_0)=H^2(\mathbb{R}^2)\) and \(A_0u=-i\Delta u\) for \(u\in D(A_0)\) for the initial value problem, we hereby established that \(A_0\) is the infinitesimal generator of a \(C_0\)-semigroup of unitary operators \(T(t)\), \(-\infty

Abstract:In this paper, closed forms of the sum formulas \(\sum\limits_{k=0}^{n}kx^{k}W_{k}\) and \(\sum\limits_{k=1}^{n}kx^{k}W_{-k}\) for generalized Tetranacci numbers are presented. As special cases, we give summation formulas of Tetranacci, Tetranacci-Lucas, and other fourth-order recurrence sequences.