Solving Fundamental Solution of Non-Homogeneous Heat Equation with Dirichlet Boundary Conditions

In this study, we developed a solution of nonhomogeneous heat equation with Dirichlet boundary conditions. moreover, the non-homogeneous heat equation with constant coefficient. since heat equation has a simple form, we would like to start from the heat equation to find the exact solution of the partial differential equation with constant coefficient. to emphasize our main results, we also consider some important way of solving of partial differential equation specially solving heat equation with Dirichlet boundary conditions. the main results of our paper are quite general in nature and yield some interesting solution of non-homogeneous heat equation with Dirichlet boundary conditions and it is used for problems of mathematical modeling and mathematical physics.

Invariants in Optimal Control: An Exact Solution of the Optimal Stabilization Problem

The stabilizing optimal feedback is a function onthe separatrix of stable points of the associated Hamiltoniansystem. Three geometric objects - the symplectic form, Hamiltonianvector field, and Lyapunov function, generating the separatrix - are %intrinsic to the optimal control system. They areinvariantly attached to the optimal control system under canonicaltransformations of the phase space. The separatrix equations can be writtenin terms of these invariants through invariant operations.There is a computable representative of the equivalence class,containing the original system. It is its linear approximation system at the stable point.

A Descriptive Time Series Analysis Applied to the Fit of Carbon-Dioxide (CO2)

The study examined the use of population spectrum in determining the nature (deterministic and stochastic) of trend and seasonal component of given time series. It also adopts the use of coefficient of variation approach in the choice of appropriate model in descriptive time series technique. Illustrations were carried out using average monthly atmospheric Carbon dioxide (C02) from 2000-2017 with 2018 used for forecast. Spectrum analysis showed that the descriptive technique of time series is more appropriate for analysis of the study data. The coefficient of variation revealed that the multiplicative model was appropriate for the CO2 data while the forecast and the actual values showed no significant mean difference at 5% level of significance.

On F-Polynomial, Multiple and Hyper F-Index of some Molecular Graphs

A graph can be recognized by a numeric number, a polynomial, a sequence of numbers or a matrix which represent the whole graph, and these representations are aimed to be uniquely defined for that graph. Topological index is a numeric quantity with a graph which characterizes the topology of the graph and is invariant under graph automorphism. In this paper, we compute F-polynomial, Multiple F-index and Hyper F-index for some special graphs.

Soliton Solutions of Space-Time Fractional-Order Modified Extended Zakharov-Kuznetsov Equation in Plasma Physics

The aim of this article is to calculate the soliton solutions of space-time fractional-order modified extended Zakharov-Kuznetsov equation which is modeled to investigate the waves in magnetized plasma physics. Fractional derivatives in the form of modified Riemann-Liouville derivatives are used. Complex fractional transformation is applied to convert the original nonlinear partial differential equation into another nonlinear ordinary differential equation. Then, soliton solutions are obtained by using (1/G')-expansion method. Bright and dark soliton solutions are also obtain with ansatz method. These solutions may be of significant importance in plasma physics where this equation is modeled for some special physical phenomenon.

Ablowitz-Kaup-Newel-Segur Formalism and N-Soliton Solutions of Generalized Shallow Water Wave Equation

We apply Ablowitz-Kaup-Newel-Segur hierarchy to derive the generalized shallow waterwave equation and we also investigate N-soliton solutions of the derived equation using InverseScattering Transform method and Hirota’s bilinear method.

Network Routing on Regular Directed Graphs from Spanning Factorizations

Networks with a high degree of symmetry are useful models for parallel processor networks. In earlier papers, we defined several global communication tasks (universal exchange, universal broadcast, universal summation) that can be critical tasks when complex algorithms are mapped to parallel machines. We showed that utilizing the symmetry can make network optimization a tractable problem. In particular, we showed that Cayley graphs have the desirable property that certain routing schemes starting from a single node can be transferred to all nodes in a way that does not introduce conflicts. In this paper, we define the concept of spanning factorizations and show that this property can also be used to transfer routing schemes from a single node to all other nodes. We show that all Cayley graphs and many (perhaps all) vertex transitive graphs have spanning factorizations.

Comparative Study of Matlab ODE Solvers for the Korakianitis and Shi Model

Changing parameters of the Korakianitis and Shi heart valve model over a cardiac cycle has led to the investigation of appropriate numerical technique(s) for good speed and accuracy. Two sets of parameters were selected for the numerical test. For the seven MATLAB ODE solvers, the computed results, computational cost and execution time were observed for varied error tolerance and initial time steps. The results were evaluated with descriptive statistics; the Pearson correlation and ANOVA at The dependence of the computed result, accuracy of the method, computational cost and execution time of all the solvers, on relative tolerance and initial time steps were ascertained. Our findings provide important information that can be useful for selecting a MATLAB ODE solver suitable for differential equation with time varying parameters and changing stiffness properties.

Extrapolation Problem for Continuous Time Periodically Correlated Isotropic Random Fields

The problem of optimal linear estimation of functionals depending on the unknown values of a random fieldζ(t,x), which is mean-square continuous periodically correlated with respect to time argumenttє R and isotropic on the unit sphere Sn with respect to spatial argumentxєSn. Estimates are based on observations of the fieldζ(t,x) +Θ(t,x) at points (t,x) :t< 0;xєSn, whereΘ(t,x) is an uncorrelated withζ(t,x) random field, which is mean-square continuous periodically correlated with respect to time argumenttє R and isotropic on the sphereSnwith respect to spatial argumentxєSn. Formulas for calculating the mean square errors and the spectral characteristics of the optimal linear estimate of functionals are derived in the case of spectral certainty where the spectral densities of the fields are exactly known. Formulas that determine the least favourable spectral densities and the minimax (robust) spectral characteristics are proposed in the case where the spectral densities are not exactly known while a class of admissible spectral densities is given.

Computing F-Index of Different Corona Products of Graphs

F-index of a graph is equal to the sum of cubes of degree of all the vertices of a given graph. Among different products of graphs, as corona product of two graphs is one of most important, in this study, the explicit expressions for F-index of different types of corona product of are obtained.