This paper studies the fourth-order problem with multi-term time fractional integral operator under simply supported type conditions. We first introduce a novel computational approach, the discrete singular convolution (DSC) algorithm, for analyzing this problem. Detailed discrete formulations and the treatment of simply supported boundary condition are established. We provide some numerical results to demonstrate the validity and applicability of the proposed technique. Comprehensive comparisons are given based on a variety of time increment, grid spacing and wave number. Unified features of the DSC algorithm for solving differential equations are explored. It is demonstrated that the DSC algorithm is an accurate, stable and robust approach for solving the fourth-order integro-differential equation with multi-term time fractional integral operator.

In this paper, we study the value sharing problems for certain types of difference polynomials with the notion of weakly weighted sharing. Our results improve and generalize the results due to P. Sahoo and B. Saha [App. Math. E-Notes. 16(2016), 33-44], H.P. Waghamore [Tbilisi Math. J. 11(2018), 1-13].

This paper aims to obtain exact and numerical solutions of the nonlinear Benjamin Bona Mahony-Burgers (BBM-Burgers) equation. Here, we propose the modified Kudryashov method for getting the exact traveling wave solutions of BBM-Burgers equation and a septic B-spline collocation finite element method for numerical investigations. The numerical method is validated by studying solitary wave motion. Linear stability analysis of the numerical scheme is done with Fourier method based on von-Neumann theory. To show suitability and robustness of the new numerical algorithm, error norms $L_{2}$, $L_{\infty }$ and three invariants $I_{1},I_{2}$ and $I_{3}$ are calculated and obtained results are given both numerically and graphically. The obtained results state that our exact and numerical schemes ensure evident and they are penetrative mathematical instruments for solving nonlinear evolution equation.

In the present paper, by virtue of probability approach, we study an open problem which was proposed by " F. Qi, Several integral inequalities, J. Inequal. Pure Appl. Math. 1 (2000), no. 2, Art. 19; available online at {http://emis.icm.edu.pl/journals/JIPAM/article113.html?sid=113}.", and obtain several generalized Qi-type inequalities which extend and improve some known results.

In this paper, introducing the notion of topologically simple pro $C^*$-algebras, we show that direct product of $C^*$-algebras $K(H_i)$, as the set of all compact operators on a Hilbert space $H_i$, is a topologically simple pro $C^*$-algebra. Applying this fact, we prove that the set of all bounded elements of a certain class of Hilbert modules are dense in the same module.

In this work, we construct vector spaces $l_{p}\left( \mathbb{BC}\left(N\right) \right) $ of absolutely $p-$ summable $\ast -$bicomplex sequences with the $\ast -$ norm $\overset{..}{\parallel }.\overset{..}{\parallel }_{2,l_{p}\left( \mathbb{BC}\left( N\right) \right) }$ over the field $\mathbb{C}\left( N\right).$ Also, we show that some inclusion relations hold and these vector spaces are Banach spaces by using Minkowski's inequality in $\mathbb{BC}\left( N\right) $ with respect to $\overset{..}{\parallel }.\overset{..}{\parallel }_{2}.$

In this research, we discuss the construction of analytic solution of non-homogenous initial boundary value problem including PDEs of fractional order. By means of separation of variables method, the solution is constructed in the form of a Fourier series with respect to the eigenfunctions of a corresponding Sturm-Liouville eigenvalue problem including fractional derivative in Liouville-Caputo sense.

In this paper, considering the concept of $f-$statistical convergence which is a generalization of statistical convergence and is intermediate between the ordinary convergence and the statistical convergence, we obtain a $f-$statistical approximation theorem for sequences of positive linear operators defined on the space of all real valued $B-$continuous functions on a compact subset of the real line. Furthermore, we compute the rates of $f-$statistical convergence with the help mixed modulus of smoothness.

In this paper, we find all Fibonacci numbers which are products of two Jacobsthal numbers. Also we find all Jacobsthal numbers which are products of two Fibonacci numbers. More generally, taking $k,m,n$ as positive integers, it is proved that $F_{k}=J_{m}J_{n}$ implies that \begin{align*} (k,m,n) = &(1,1,1),(2,1,1),(1,1,2),(2,1,2),\\ & (1,2,2),(2,2,2),(4,1,3),(4,2,3),\\ & (5,1,4),(5,2,4),(10,4,5),(8,1,6),(8,2,6) \end{align*} and $J_{k}=F_{m}F_{n}$ implies that \begin{align*} (k,m,n) =&(1,1,1),(2,1,1),(1,2,1),(2,2,1),\\ & (1,2,2),(2,2,2),(3,4,1),(3,4,2),\\ & (4,5,1),(4,5,2),(6,8,1),(6,8,2). \end{align*}

In this paper, we investigate the existence of solution for not instantaneous impulsive fractional differential equations with random parameter. By using fixed point theorems, we establish sufficient conditions for the existence and uniqueness of solutions. Finally, generalized Ulam Hyers Rassias stable solution is also obtained.

After the recent works of Biswas [19], on the idea of relative $(p, q)$-th order and relative $(p, q)$-th type of composite $p$-adic entire functions, in this paper we establish some results of growth properties of an $p$-adic entire functions of several complex variables with respect to another $p$-adic entire function of several complex variables $g\in \mathcal{A}(\mathbb{K})$ on the basis of their relative $(p, q)$-th order, relative $(p, q)$-th lower order, relative $(p,q)$-th type, relative $(p,q)$-th type of an entire functions of several complex variables.

In this paper we introduce $\zeta -$contraction via simulation function in uniform spaces and prove new fixed point theorem for $\zeta-$contraction in uniform spaces.

In this paper, we introduce a new class of Legendre poly-Genocchi polynomials and give some identities of these polynomials related to the Stirling numbers of the second kind. The concept of poly-Bernoulli numbers $B_{n}^{(k)}(a,b)$, poly-Bernoulli polynomials $B_{n}^{(k)}(x,a,b)$ of Jolany et al., Hermite-Bernoulli polynomials ${}_{H}B_{n}(x,y)$ of Dattoli et al., ${}_{H}B_{n}^{(\alpha)}(x,y)$ of Pathan et al. and ${}_{H}G_{n}^{(k)}(x,y)$ of Khan are generalized to the one $_{S}G_{n}^{(k)}(x,y,z)$. Some implicit summation formulae and general symmetry identities are derived by using different analytical means and applying generating function. These results extended some known summation and identities of Hermite poly-Genocchi numbers and polynomials.

In this work, we consider that the two-mode equations, especially two-mode Korteweg-de Vries equation and two-mode Sharma-Tasso-Olver equations, which are used for modelling of shallow water waves, electromagnetism, electrical and electronics engineering, signal analysis, quantum mechanics etc. and their analytical solutions are obtained via the well-known Bernoulli equation method through the symbolic computation.

In the present paper we consider the generalized Bernstein-Chlodowsky polynomial and two variable generalized Szasz operator. For the first operator we obtain convergence property on bounded intervals of positive semi-axis. For the second operator, we obtain direct approximation property on positive semi-axis as estimate of the two variable generalized Szasz operator by two variable weighted modulus of continuity.

This paper is concerned with the existence and nonexistence of positive solutions for a system of nonlinear second order dynamic equations with multi-point boundary conditions on time scales.

In this paper we establish a new Wilker type and Huygens type inequalities involving the trigonometric and hyperbolic functions. Moreover, in terms of hyperbolic functions, the upper and lower bounds of $\sin(x)/x$ and $\tan(x)/x$ are given.

In this study, we introduce a new family of generating functions of products of bivariate complex Fibonacci polynomials with Gaussian Fibonacci numbers, Gaussian Lucas numbers, Gaussian Jacobsthal numbers, Gaussian Jacobsthal Lucas numbers, Gaussian Pell numbers and Gaussian Pell Lucas numbers.

In this article, the author used Mellin and Kontorovich-Lebedev transforms to establish certain integrals involving Macdonald's functions. Transform method is a powerful tool for solving singular integral equations, evaluation of certain integrals and solution to partial differential equations. The result reveals that the transform method is very convenient and effective.

In this paper, some new estimates of Hermite-Hadamard-Fejér inequality for $n$-times differentiable preinvex functions are established. It is noteworthy that the weighted estimates of the left and right hand side of the Hermite-Hadamard inequalities involving generalized preinvex functions doesn't exist previously.

It is shown that the following $\left( k+l\right) $-order nonlinear difference equation $$x_{n}=\frac{x_{n-k}x_{n-k-l}}{x_{n-l}\left( a_{n}+b_{n}x_{n-k}x_{n-k-l}\right)}, \ n\in \mathbb{N}_{0},$$ where $k,l\in \mathbb{N}$, $\left(a_{n} \right)_{n\in \mathbb{N}_{0}}$, $\left(b_{n} \right)_{n\in \mathbb{N}_{0}}$ and the initial values $x_{-i}$, $i=\overline {1,k+l}$, are real numbers, can be solved and extended some results in literature. Also, by using obtained formulas, we give the forbidden set of the initial values for aforementioned equation and study the asymptotic behavior of well-defined solutions of above difference equation for the case $k=3$, $l=k$.