On applicability of truncation method for damped axially moving string

In this paper, the detailed study of the transversal vibrations of a damped axially moving string is considered. Two end pulleys of the string are taken to be fixed and the initial conditions are assumed to be of general displacement field and the general velocity field. The axial speed of the string is considered to be sinusoidal, time-dependent and small compared to wave-velocity. A two timescales perturbation method with a combination of Fourier-sine series which fits the boundary conditions is employed in order to formulate the valid and uniform asymptotic approximations of the exact solutions for the equation. It is found that there are infinitely many values of frequency parameter Ω which cause the internal resonances in system. The fundamental resonant frequency, the non-resonant frequency and the detuning cases have been discussed and analyzed in detail. It has been found explicitly that the total mechanical energy of the infinite dimensional system decreases for two cases of the damping parameter, that is, for δ=2 and for δ>2. By truncation method it has been shown that the mode-amplitude response for first few modes is stable. So, Galerkin’s truncation method may be possible for these two cases of the parameter δ. But for case δ<2 the total mechanical energy of belt system is increasing exponentially. Therefore, it is evident that the Galerkin’s truncation method cannot be applied in order to obtain valid approximations on long timescales, that is, on timescales of O1/ε

A Novel Two Point Optimal Derivative free Method for Numerical Solution of Nonlinear Algebraic, Transcendental Equations and Application Problems using Weight Function

It’s a big challenge for researchers to locate the root of nonlinear equations with minimum cost, lot of methods are already exist in  literature to find root but their cost are very high In this regard we introduce a two-step  fourth order method by using weight function. And proposed method is optimal and derivative free for solution of nonlinear algebraic and transcendental and application problems. MATLAB, Mathematica and Maple software are used to solve the convergence and numerical problems of proposed and their counterpart methods

Exact Solutions on the Oscillating Plate of Maxwell Fluids

This work is related to establish the exact solutions of sine hyperbolic and cosine hyperbolic oscillations of Maxwell fluid over the velocity field and shear stress. Under the effects of sine hyperbolic and cosine hyperbolic oscillations, the general solutions are derived for the motions of incompressible Maxwell fluid. For the sack of the general solutions the mathematical techniques of integral transformations (Laplace and Fourier Sine transforms) are applied. We have expressed the obtained solutions under form of theorem of convolutions product and integral notation, satisfying the boundary and initial conditions. The expressions for similar solutions are specialized as a limiting case of Newtonian fluid

Oscillating Flows of Fractionalized Second Grade Fluid with Slip Effects

This paper examines the fractionalized second grade fluid dueto oscillating plate under slip condition. The discrete Laplace transformtechnique is employed to compute the analytical solutions for the equa-tions of motion. The velocity field and shear stress are computed. In orderto write them in compact form, the Wright generalized hyper geometricfunction is used and written as addition of slip and no slip contributions.The closed-form solutions for ordinary second grade and Newtonian flu-ids carrying out the similar motion are achieved. The computations forfractional and ordinary second grade fluids without slip effect are alsoachieved as a special case. Furthermore, the impact of various parameterssuch as the slip, fractional and material parameters on the motion of frac-tionalized second grade fluid will be explained through graphs. Finally,a comparison among the Newtonian fluids, ordinary second grade fluidsand fractionalized second grade fluids is also carried out

Facile synthesis and characterization of beta-Cd(OH)(2) nanostructures for adsorptive removal of Cr(VI) ions from wastewater: a statistical approach for multivariate sorption optimization

In the present study, nanostructured beta-Cd(OH)(2) adsorbent was synthesized, characterized by Fourier-transform infrared spectroscopy, X-ray diffraction and scanning electron microscopy analysis, and applied for Cr(VI) ions capturing (adsorption) from environmental aqueous samples. The central composite design of 18 adsorption experiments was employed for multivariate sorption optimization. Maximum adsorption (%) of Cr(VI) ions was calculated and found to be 98.5% with relative standard deviation (RSD) = 3.5 at optimum concentration 15 mg L-1, pH 4.0, adsorbent dosage 50 mg, shaking time 20 min and shaking speed 120 rpm at 25 degrees C. Langmuir, Freundlich and Dubinin-Radushkevich isotherms fitted well to adsorption data with correlation coefficient (R-2) of 0.993, 0.982 and 0.994, respectively. Mono-layered (Q(m)) and multi-layered (K-f) capacities of beta-Cd(OH)(2) adsorbent for Cr( VI) ions retention were calculated and found to be 202.02 +/- 2.0 and 4.95 +/- 2.5 mg g(-1), respectively. Sorption energy was calculated and found to be 8.45 +/- 2.0 kJ mol(-1), indicated chemisorption or ion exchange mechanism for Cr(VI) ions adsorption onto beta-Cd(OH)(2) adsorbent

Characterizations of Chemical Networks Entropies by <i>K</i>-Banhatii Topological Indices

Entropy is a thermodynamic function in physics that measures the randomness and disorder of molecules in a particular system or process based on the diversity of configurations that molecules might take. Distance-based entropy is used to address a wide range of problems in the domains of mathematics, biology, chemical graph theory, organic and inorganic chemistry, and other disciplines. We explain the basic applications of distance-based entropy to chemical phenomena. These applications include signal processing, structural studies on crystals, molecular ensembles, and quantifying the chemical and electrical structures of molecules. In this study, we examine the characterisation of polyphenylenes and boron (B12) using a line of symmetry. Our ability to quickly ascertain the valences of each atom, and the total number of atom bonds is made possible by the symmetrical chemical structures of polyphenylenes and boron B12. By constructing these structures with degree-based indices, namely the K Banhatti indices, ReZG1-index, ReZG2-index, and ReZG3-index, we are able to determine their respective entropies

Entropy Related to K-Banhatti Indices via Valency Based on the Presence of C6H6 in Various Molecules

Entropy is a measure of a system&rsquo;s molecular disorder or unpredictability since work is produced by organized molecular motion. Shannon&rsquo;s entropy metric is applied to represent a random graph&rsquo;s variability. Entropy is a thermodynamic function in physics that, based on the variety of possible configurations for molecules to take, describes the randomness and disorder of molecules in a given system or process. Numerous issues in the fields of mathematics, biology, chemical graph theory, organic and inorganic chemistry, and other disciplines are resolved using distance-based entropy. These applications cover quantifying molecules&rsquo; chemical and electrical structures, signal processing, structural investigations on crystals, and molecular ensembles. In this paper, we look at K-Banhatti entropies using K-Banhatti indices for C6H6 embedded in different chemical networks. Our goal is to investigate the valency-based molecular invariants and K-Banhatti entropies for three chemical networks: the circumnaphthalene (CNBn), the honeycomb (HBn), and the pyrene (PYn). In order to reach conclusions, we apply the method of atom-bond partitioning based on valences, which is an application of spectral graph theory. We obtain the precise values of the first K-Banhatti entropy, the second K-Banhatti entropy, the first hyper K-Banhatti entropy, and the second hyper K-Banhatti entropy for the three chemical networks in the main results and conclusion

Some Novel Results Involving Prototypical Computation of Zagreb Polynomials and Indices for SiO4 Embedded in a Chain of Silicates

A topological index as a graph parameter was obtained mathematically from the graph&rsquo;s topological structure. These indices are useful for measuring the various chemical characteristics of chemical compounds in the chemical graph theory. The number of atoms that surround an atom in the molecular structure of a chemical compound determines its valency. A significant number of valency-based molecular invariants have been proposed, which connect various physicochemical aspects of chemical compounds, such as vapour pressure, stability, elastic energy, and numerous others. Molecules are linked with numerical values in a molecular network, and topological indices are a term for these values. In theoretical chemistry, topological indices are frequently used to simulate the physicochemical characteristics of chemical molecules. Zagreb indices are commonly employed by mathematicians to determine the strain energy, melting point, boiling temperature, distortion, and stability of a chemical compound. The purpose of this study is to look at valency-based molecular invariants for SiO4 embedded in a silicate chain under various conditions. To obtain the outcomes, the approach of atom&ndash;bond partitioning according to atom valences was applied by using the application of spectral graph theory, and we obtained different tables of atom&mdash;bond partitions of SiO4. We obtained exact values of valency-based molecular invariants, notably the first Zagreb, the second Zagreb, the hyper-Zagreb, the modified Zagreb, the enhanced Zagreb, and the redefined Zagreb (first, second, and third). We also provide a graphical depiction of the results that explains the reliance of topological indices on the specified polynomial structure parameters