An Implicit Numerical Approach for 2D Rayleigh Stokes Problem for a Heated Generalized Second Grade Fluid with Fractional Derivative

In this research work, our aim is to use the fast algorithm to solve the Rayleigh–Stokes problem for heated generalized second-grade fluid (RSP-HGSGF) involving Riemann–Liouville time fractional derivative. We suggest the modified implicit scheme formulated in the Riemann–Liouville integral sense and the scheme can be applied to the fractional RSP-HGSGF. Numerical experiments will be conducted, to show that the scheme is stress-free to implement, and the outcomes reveal the ideal execution of the suggested technique. The Fourier series will be used to examine the proposed scheme stability and convergence. The technique is stable, and the approximation solution converges to the exact result. To demonstrate the applicability and viability of the suggested strategy, a numerical demonstration will be provided.

Generalized Planar Turán Numbers

In a generalized Turán problem, we are given graphs $H$ and $F$ and seek to maximize the number of copies of $H$ in an $n$-vertex graph not containing $F$ as a subgraph. We consider generalized Turán problems where the host graph is planar. In particular, we obtain the order of magnitude of the maximum number of copies of a fixed tree in a planar graph containing no even cycle of length at most $2\ell$, for all $\ell$, $\ell \geqslant 1$. We also determine the order of magnitude of the maximum number of cycles of a given length in a planar $C_4$-free graph. An exact result is given for the maximum number of $5$-cycles in a $C_4$-free planar graph. Multiple conjectures are also introduced.

FACTORS RELATED TO NUTRITIONAL STATUS IN PRE-SCHOOL CHILDREN

Background: The problem of nutrition in children at Indonesia was still an unresolved problem. Unicef said that one in ten children were underweighted and a fifth of children were overweighted or obesed. This number will not decrease and it will increase if we not handled this problem seriously. This study was conducted with the aimed of analyzed the factors that influenced nutritional status in pre-school children, so that more effective prevention and treatment can be carried out. Method: This research was a cross sectional study with a total sample of 150 preschool child in the age of 3 to 6 years old and the parents with online questionnaire. Sampling was done by cluster random sampling. The study was conducted in PAUD at Tanah Kali Kedinding, Kenjeran, Surabaya. The data was analyzed with Spearmean test, Chi-square, and Fisher exact. Result: Knowledge p value = 0,025 r = 0,182. Income p value < 0,001 r = 0,368. Eating frequency p value < 0,001 r = 0,721. The amount of food p value < 0,001 r = 0,738. The kind of food p value < 0,001 C = 0,443. Physical activity p value = 0,438. Conclusion: Parental knowledge and income, and children's eating patterns were related to the nutritional status of pre-school children, while the physical activity of children does not affected the nutritional status of children.

Optimizing the First-Passage Process on a Class of Fractal Scale-Free Trees

First-passage processes on fractals are of particular importance since fractals are ubiquitous in nature, and first-passage processes are fundamental dynamic processes that have wide applications. The global mean first-passage time (GMFPT), which is the expected time for a walker (or a particle) to first reach the given target site while the probability distribution for the position of target site is uniform, is a useful indicator for the transport efficiency of the whole network. The smaller the GMFPT, the faster the mass is transported on the network. In this work, we consider the first-passage process on a class of fractal scale-free trees (FSTs), aiming at speeding up the first-passage process on the FSTs. Firstly, we analyze the global mean first-passage time (GMFPT) for unbiased random walks on the FSTs. Then we introduce proper weight, dominated by a parameter w(w>0), to each edge of the FSTs and construct a biased random walks strategy based on these weights. Next, we analytically evaluated the GMFPT for biased random walks on the FSTs. The exact results of the GMFPT for unbiased and biased random walks on the FSTs are both obtained. Finally, we view the GMFPT as a function of parameter w and find the point where the GMFPT achieves its minimum. The exact result is obtained and a way to optimize and speed up the first-passage process on the FSTs is presented.

ANALYSIS OF TIME-FRACTIONAL BURGERS AND DIFFUSION EQUATIONS BY USING MODIFIED q-HATM

In this paper, the q-homotopy analysis transform technique is implemented to analyze the solution of fractional-order Burgers and diffusion equations with the help of Caputo operator. The results of the proposed method are shown and analyzed with the help of figures. This approach is used to determine the solution in a convergent sequence and illustrate the q-homotopy analysis transform technique solutions convergence to the exact result. Several examples showed the reliability and simplicity of the technique and highlighted the significance of this work. Therefore, the proposed method is successful in investigating other fractional-order linear and nonlinear partial differential equations.

A SOLITARY CONVERGENT PERIODIC SOLUTION OF THE INVERSE TRULY NONLINEAR OSCILLATOR BY MODIFIED MICKENS’ EXTENDED ITERATION PROCEDURE

The inverse truly nonlinear oscillator is the most applied in the field of computer science, information technology, physics, electrical engineering, and mechanical engineering. The solution of the inverse truly nonlinear oscillator has been obtained by modified Mickens’ extended iteration procedure. To determine the solution of the oscillator a special type of Fourier series has been used. The iterated solutions are convergent as the second, third, and fourth approximate frequencies of the oscillator show a good concurrence with the exact result. Some researchers presented the solutions of the same oscillator by applying different methods. We have compared the obtained results with some previously published results. Some of their techniques diverge at higher-order stages but the present technique is convergent there. The method is mainly illustrated in the strongly nonlinear inverse oscillator, but it can be widely applicable in other problems arising from nonlinear sciences and engineering.

Initial logarithmic coefficients for functions starlike with respect to symmetric points

In this paper, we obtain the bounds of the initial logarithmic coefficients for functions in the classes $${\mathcal {S}}_S^*$$ S S ∗ and $${\mathcal {K}}_S$$ K S of functions which are starlike with respect to symmetric points and convex with respect to symmetric points, respectively. In our research, we use a different approach than the usual one in which the coeffcients of f are expressed by the corresponding coeffcients of functions with positive real part. In what follows, we express the coeffcients of f in $${\mathcal {S}}_S^*$$ S S ∗ and $${\mathcal {K}}_S$$ K S by the corresponding coeffcients of Schwarz functions. In the proofs, we apply some inequalities for these functions obtained by Prokhorov and Szynal, by Carlson and by Efraimidis. This approach offers a additional benefit. In many cases, it is easily possible to predict the exact result and to select extremal functions. It is the case for $${\mathcal {S}}_S^*$$ S S ∗ and $${\mathcal {K}}_S$$ K S .

Degeneracy Effects of Energy loss Straggling of Homo and Hetero Di-Cluster Ions

The energy loss straggling is obtained from an exact quantum mechanical evaluation, which takes into account the degeneracy of the target plasma, and later it is compared with common classical and degeneracy approximation as a function of incident Homo (H-H, He-He) and Hetero (He-H) di-cluster energy in Kev with different kinds of plasma target. For homonuclear di-clusters (H-H) and (He-He) decreasing temperature, the exact calculation approaches the high degeneracy limit, but the differences are still significant. However, as the temperature rises, the exact result approaches the classical limit. Finally, the energy loss straggling increases with the increasing atomic number of the projectiles (He-He). Our research focuses on targets in the weakly coupled electron gas limit, where we can use the random phase approximation (RPA). This kind of plasma has not been widely researched, considering the fact that it is essential for inertial confinement fusion (ICF).

NUMERICAL DIFFERENTIATION OF BEND FORMS OF THE LONG ELASTIC RODS

The technique of numerical differentiation of the bend forms of long elastic rods is presented. This technique is based on search for new bend forms of the rod by solving the equations of oscillations with using the time integration method and the polynomial spline-functions that are being described the current bend form. In it, the spline-functions are found by current bend form approximation where each of the found functions is responsible to certain point of rod elastic line and describes the position of nearby points. Using the described approximation technique with subsequent numerical differentiation, the dependences of the derivatives on an arbitrary bend form of the rod with a length that is equal to 100 m are shown. To confirm the reliability, the results of numerical differentiation of the bend forms of the elastic rods described by given functions are presented and the numerical results obtained using the proposed method are compared with the results of analytical differentiation of the original functions. The graphs of values derivatives dependence to rod length are drawn and tables with numerical values of differentiation results are shown. It is concluded that the considered technique of numerical differentiation of rods bend forms allows to do the research of dynamics of rod systems. It gives the exact result of differentiation, provides the continuity and smoothness of all four derivatives functions of spline that are being described the bend form with considerable length. Described technique was realized in a computer program with graphic user interface. Program allows to monitor for dynamics of the oscillatory motion of the modeled system in real-time by calculating and drawing the current band forms of the rotating rod during the oscillation.