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.

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.

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.

Closed-form expressions are established for dimensionless long-tome solutions of some mixed initial-boundary value problems. They correspond to three isothermal unsteady motions of a class of incompressible Maxwell fluids with power-law dependence of viscosity on the pressure. The fluid motion, between infinite horizontal parallel flat plates, is induced by the lower plate that applies time-dependent shear stresses to the fluid. As a check of the obtained results, the similar solutions corresponding to the classical incompressible Maxwell fluids performing same motions are recovered as limiting cases of present solutions. Finally, some characteristics of fluid motion as well as the influence of pressure-viscosity coefficient on the fluid motion are graphically presented and discussed.

Let \(S\) be a dominating set of a graph \(G\). The set \(S\) is called a pendant dominating set of \(G\) if the induced subgraph of \(S\) contains a minimum of one node of degree one. The minimum cardinality of the pendant dominating set in \(G\) is referred to as the pendant domination number of \(G\), indicated by \(\gamma_{pe}(G)\). This article analyzes the effect of \(\gamma_{pe}(G)\) when an arbitrary node or edge is removed.

In this paper, we have proposed new windmill graph, that is Basava wheel windmill graph. The Basava wheel windmill graph \(W^{(m)}_{n+1}\) is the graph obtained by taking \(m\geq 2\) copies of the graph \(K_1+W_{n}\) for \(n\geq 4\) with a vertex \(K_1\) in common. Inspired by recent work on topological indices, proposed new degree-based topological indices namely, general \(SK_{\alpha}\) and \(SK^{\alpha}_1\) indices of a graph \(G\). We have obtained first and second Zagreb index, F-index, first and second hyper-Zagreb index, harmonic index, Randi\(\acute{c}\) index, general Randi\(\acute{c}\) index, sum connectivity index, general sum connectivity index, atom-bond connectivity index, geometric-arithmetic index, Symmetric division deg index, Sombor index, SK indices, general \(SK_{\alpha}\) and \(SK^{\alpha}_1\) indices of Basava wheel windmill graph. Further, we have computed exact values of these topological indices of chloroquine, hydroxychloroquine and remdesiver.

Our primary purpose is to compute explicitly traces of the Dirichlet forms related to Feller’s one-dimensional diffusions on countable sets via Fukushima’s method. For discrete measures, the obtained trace form can be described as a Dirichlet form on the graph.

In this paper, closed forms of the sum formulas \(\sum\limits_{k=0}^{n}x^{k}W_{mk+j}^{3}\) for generalized balancing numbers are presented. As special cases, we give sum formulas of balancing, modified Lucas-balancing and Lucas-balancing numbers.

We construct a class of quadratic irrationals having continued fractions of period \(n\geq2\) with `small’ partial quotients for which specific integer multiples have periodic continued fractions with the length of the period being \(1\), \(2\) or \(4\), and with ‘large’ partial quotients. We then show that numbers in the period of the new continued fraction are functions of the numbers in the periods of the original continued fraction. We also show how polynomials arising from generalizations of these continued fractions are related to Chebyshev and Fibonacci polynomials and, in some cases, have hyperbolic root distribution.

In this study, we focus on the slip effects on the peristaltic unsteady flow of magnatohydromagnetic Jeffrey fluid in a flow passage with non-conducting and flexible boundary walls. The effect of the magnetic field with varying thermal conductivity is taken under the influence of heat transfer analysis. The dimensionless system of PDEs is solved analytically, and the obtained results are computed for the temperature, pressure drop, the axial pressure gradient, axial velocity, and then these results are discussed for different values of the physical parameters of our interest. For the stream functions, the contour plots are also obtained which indicates the exact flow behavior within the flow channel, and the effects of the physical parameters on Jeffery fluid within the flow channel are discussed briefly. Our results indicate that the heat transfer coefficient decreases with an increase in thermal slip and velocity slip parameters. Furthermore, it shows that the size of the trapped bolus is greater for the inclined magnetic field as compared to the transverse magnetic field.