Open Journal of Mathematical Analysis (OMA)

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.

Abstract:This work is devoted to study the global solutions of a class of nonlinear Moore-Gibson-Thompson equation. By applying the Galerkin and compact methods, we derive some sufficient conditions on the nonlinear terms, which lead to the existence and uniqueness of the global solution.

Abstract:This work is an in-depth study of the class of norm-attainable operators in a general Banach space setting. We give characterizations of norm-attainable operators on involutive stereotype tubes with algebraically connected component of the identity. In particular, we prove reflexivity, boundedness and compactness properties when the set of these operators contains unit balls with involution for the tubes when they are of stereotype category.

Abstract:In this article we study the existence of periodic and strong solutions of Navier-Stokes equations, in two dimensions, with non-local viscosity.

Abstract:In this paper we consider the following abstract class of weakly dissipative second-order systems with infinite memory, \(u”(t)+Au(t)-\displaystyle\int_{0}^{\infty} g(s)A^\alpha u(t-s)ds=0,~t>0,\) and establish a general stability result with a very general assumption on the behavior of \(g\) at infinity; that is \(g'(t) \leq – \xi(t) G \left(g(t)\right),~~t \geq 0.\) where \(\xi\) and \(G\) are two functions satisfying some specific conditions. Our result generalizes and improves many earlier results in the literature. Moreover, we obtain our result with imposing a weaker restrictive assumption on the boundedness of initial data used in many earlier papers in the literature such as the one in [1-5]. The proof is based on the energy method together with convexity arguments.

Abstract:In this paper, we study the initial boundary value problem for a p-biharmonic parabolic equation with logarithmic nonlinearity. By using the potential wells method and logarithmic Sobolev inequality, we obtain the existence of the unique global weak solution. In addition, we also obtain decay polynomially of solutions.

Abstract:In this paper, we investigate some properties of the \(AP\)-Henstock integral on a compact set and prove that the product of an \(AP\)-Henstock integrable function and a function of bounded variation is \(AP\)-Henstock integrable. Furthermore, we prove that the product of an \(AP\)-Henstock integrable function and a regulated function is also \(AP\)-Henstock integrable. We also define the \(AP\)-Henstock integral on an unbounded interval, investigate some properties, and show similar multiplier properties.

Abstract:This paper consists of the results about \(\omega\)-order preserving partial contraction mapping using perturbation theory to generate a one-parameter semigroup. We show that adding a bounded linear operator \(B\) to an infinitesimal generator \(A\) of a semigroup of the linear operator does not destroy A’s property. Furthermore, \(A\) is the generator of a one-parameter semigroup, and \(B\) is a small perturbation so that \(A+B\) is also the generator of a one-parameter semigroup.

Abstract:We consider 1D and 2D Schrödinger equation with delta potential on the positive half-axis with Dirichlet, Neumann, and Robin type boundary conditions. We presented and estimated the exact values of the beta critical.

Abstract:Two terminating balanced \(_4\phi_3\)-series identities are established by applying the bilateral \(q\)-Legendre inversions. Four variants of them are obtained by means of contiguous relations. According to the polynomial argument, four “dual” formulae for balanced \(_4\phi_3\)-series are deduced, that lead also to four non-terminating \(_2\phi_2\)-series identities.

Abstract:The martingale analogue of Kolmogorov’s law of the iterated logarithm was obtained by W. Stout using probabilistic approach. In this paper, we give a new proof of one side of the same law of the iterated logarithm for dyadic martingale using subgaussian type estimates and Borel-Cantelli Lemma.