International Science Index

International Journal of Mathematical and Computational Sciences

Application of Mathematical Sciences to Farm Management
Agriculture has been the mainstay of the nation’s economy in Nigeria. It provides food for the ever rapidly increasing population and raw materials for the industries. People especially the rural dwellers are gainfully employed on their crop farms and small-scale livestock farms for income earning. In farming, availability of funds and time management are one of the major factors that influence the system of farming in Nigeria in which mathematical science knowledge was highly required in order for farms to be managed effectively. Farmers often applied mathematics, almost every day for a variety of tasks, ranging from measuring and weighing, to land marking. This paper, therefore, explores some of the ways math is used in farming. For instance, farmers use arithmetic variety of farm activities such as seed planting, harvesting crop, cultivation and mulching. It is also important in helping farmers to know how much their livestock weighs, how much milk their cows produce and crop yield per acres, among others.
An Equivalence Between A Harmonic Form And A Closed Co-Closed Form In L^Q and Non-L^Q Spaces
An equivalent relation between a harmonic form and a closed co-closed form is established on a complete non-compact manifold. This equivalence has been generalized for a differential k-form ω from Lq spaces to non-Lq spaces when q=2 in the context of p-balanced growth where p=2. Especially for a simple differential k-form on a complete non-compact manifold, the equivalent relation has been verified with the extended scope of q for from finite q-energy in Lq spaces to infinite q-energy in non-Lq spaces when with 2-balanced growth. Generalized Hadamard Theorem, Cauchy-Schwarz Inequality, and Calculus skills including Integration by Parts as well as Convergent Series have been applied as estimation techniques to evaluate growth rates for a differential form. In particular, energy growth rates as indicated by an appropriate power range in a selected test function lead to a balance between a harmonic differential form and a closed co-closed differential form. Research ideas and computational methods in this paper could provide an innovative way in the study of broadening Lq spaces to non-Lq spaces with a wide variety of infinite energy growth for a differential form.
Applying p-Balanced Energy Technique to Solve Liouville-Type Problems in Calculus
The study of Liouville-type problems in Differential Geometry is to discover constancy properties for maps between Riemannian manifolds. Existence of constancy properties is determined by geometric structures on manifolds and energy growth for maps. In this article, we focus on solving Liouville-type problems in Calculus where manifolds are restricted with the real number systems and maps become functions. Calculus skills such as Holder Inequality and Tests for Series will be used to evaluate limits and integrations for function energy in calculation. Liouville-type result of vanishing properties for functions with appropriate energy growth is obtained. The original work in our research findings is to extend the q-energy growth from finite Lᑫ space to infinite non-Lᑫ space in the context of p-balanced energy growth where q=p=2. An algorithm and energy estimation techniques for functions in this article can be generalized as a successfully computational method to apply the technique of p-balanced energy growth for maps between manifolds as an extension of q-energy from finite to infinite.
Wavelet Method for Numerical Solution of Fourth Order Wave Equation
In this paper, a highly accurate numerical method for the solution of one-dimensional fourth-order wave equation is derived. This hyperbolic problem is solved by using semi-discrete approximations. The space direction is discretized by Wavelet-Galerkin Method and the time variable is discretized by using Newmark schemes. The numerical results show that this method gives high favourable accuracy while compared to the exact solution.
Trinary Affinity—Mathematic Verification and Application (1): Construction of Formulas for the Composite and Prime Numbers
Trinary affinity is a description of existence: every object exists as it is known and spoken of, in a system of 2 differences (denoted dif1, dif₂) and 1 similarity (Sim), equivalently expressed as dif₁ / Sim / dif₂ and kn / 0 / tkn (kn = the known, tkn = the 'to be known', 0 = the zero point of knowing). They are mathematically verified and illustrated in this paper by the arrangement of all integers onto 3 columns, where each number exists as a difference in relation to another number as another difference, and the 2 difs as arbitrated by a third number as the Sim, resulting in a trinary affinity or trinity of 3 numbers, of which one is the known, the other the 'to be known', and the third the zero (0) from which both the kn and tkn are measured and specified. Consequently, any number is horizontally specified either as 3n, or as '3n – 1' or '3n + 1', and vertically as 'Cn + c', so that any number seems to occur at the intersection of its X and Y axes and represented by its X and Y coordinates, as any point on Earth’s surface by its latitude and longitude. Technically, i) primes are viewed and treated as progenitors, and composites as descending from them, forming families of composites, each capable of being measured and specified from its own zero called in this paper the realistic zero (denoted 0r, as contrasted to the mathematic zero, 0m), which corresponds to the constant c, and the nature of which separates the composite and prime numbers, and ii) any number is considered as having a magnitude as well as a position, so that a number is verified as a prime first by referring to its descriptive formula and then by making sure that no composite number can possibly occur on its position, by dividing it with factors provided by the composite number formulas. The paper consists of 3 parts: 1) a brief explanation of the trinary affinity of things, 2) the 8 formulas that represent ALL the primes, and 3) families of composite numbers, each represented by a formula. A composite number family is described as 3n + f₁‧f₂. Since there are an infinitely large number of composite number families, to verify the primality of a great probable prime, we have to have it divided with several or many a f₁ from a range of composite number formulas, a procedure that is as laborious as it is the surest way to verifying a great number’s primality. (So, it is possible to substitute planned division for trial division.)
Time Series Modelling and Predict of River Runoff Case Study: Karkheh River at Iran
Rainfall and runoff phenomenon is a chaotic and complex outcome of nature which requires sophisticated modelling and simulation methods for explanation and use. Time Series Modelling allows runoff data analysis and can be used as forecasting tool. In the paper, attempt is made to model river runoff data and predict the future behavioral pattern of river based on annual past observations of annual river runoff. The river runoff analysis and predict are done using ARIMA model. For evaluating the efficiency of prediction to hydrological events such as rainfall, runoff and etc., we use the statistical formulae applicable. The good agreement between predicted and observation river runoff coefficient of determination (R²) display that the ARIMA (4,1,1) is the suitable model for predicting Karkheh river runoff at Iran.
On the Bootstrap P-Value Method in Identifying out of Control Signals in Multivariate Control Chart
In any production process, every product is aimed to attain a certain standard, but the presence of assignable cause of variability affects our process thereby leading to low quality of product. The ability to identify and remove this type of variability reduces its overall effect thereby improving the quality of the product. When a univariate control chart signal, it is easy to detect the problem and give a solution since it is related to a single quality characteristic. However, the problems involved in the use of multivariate control chart are the violation of multivariate normal assumption and the difficulty in identifying the quality characteristic(s) that resulted in the out of control signals. The purpose of this paper is to examine the use of non-parametric control chart (the bootstrap approach) for obtaining control limit to overcome the problem of multivariate distributional assumption and the p-value method for detecting out of control signals. Results from a performance study shows that the proposed bootstrap method enables the setting of control limit that can enhance the detection of out of control signals when compared, while the p-value method also enhanced in identifying out of control variables.
A Fundamental Functional Equation for Lie Algebras
Inspired by the so called Jacobi Identity (x y) z + (y z) x + (z x) y = 0, the following class of functional equations EQ I: F [F (x, y), z] + F [F (y, z), x] + F [F (z, x), y] = 0 is proposed, researched and generalized. Research methodologies begin with classical methods for functional equations, then evolve into discovering of any implicit algebraic structures. One of this paper’s major findings is that EQ I, under two additional conditions F (x, x) = 0 and F (x, y) + F (y, x) = 0, proves to be a fundamental functional equation for Lie Algebras. Existence of non-trivial solutions for EQ I can be proven by defining F (p, q) = [p q] = pq –qp, where p and q are quaternions, and pq is the quaternion product of p and q. EQ I can be generalized to the following class of functional equations EQ II: F [G (x, y), z] + F [G (y, z), x] + F [G (z, x), y] = 0. Concluding Statement: With a major finding proven, and non-trivial solutions derived, this research paper illustrates and provides a new functional equation scheme for studies in two major areas: (1) What underlying algebraic structures can be defined and/or derived from EQ I or EQ II? (2) What conditions can be imposed so that conditional general solutions to EQ I and EQ II can be found, investigated and applied?
Using Convergent and Divergent Thinking in Creative Problem Solving in Mathematics
This paper aims to find out how students using convergent and divergent thinking in creative problem solving to solve mathematical problems creatively. Eight engineering undergraduates in a local university took part in this study. They were divided into two groups. They solved the mathematical problems with the use of creative problem solving skills. Their solutions were collected and analyzed to reveal all the processes of problem solving, namely: problem definition, ideas generation, ideas evaluation, ideas judgment, and solution implementation. The result showed that the students were able to solve the mathematical problem with the use of creative problem solving skills.
Curve Fitting by Cubic Bezier Curves Using Migrating Birds Optimization Algorithm
A new met heuristic optimization algorithm called as Migrating Birds Optimization is used for curve fitting by rational cubic Bezier Curves. This requires solving a complicated multivariate optimization problem. In this study, the solution of this optimization problem is achieved by Migrating Birds Optimization algorithm that is a powerful met heuristic nature-inspired algorithm well appropriate for optimization. The results of this study show that the proposed method performs very well and being able to fit the data points to cubic Bezier Curves with a high degree of accuracy.
Analysis of an Error Estimate for the Asymptotic Solution of the Heat Conduction Problem in a Dilated Pipe
Subject of this study is the stationary heat conduction problem through a pipe filled with incompressible viscous fluid. In previous work, we observed the existence and uniqueness theorems for the corresponding boundary-value problem and within we have taken into account the effects of the pipe's dilatation due to the temperature of the fluid inside of the pipe. The main difficulty comes from the fact that flow domain changes depending on the solution of the observed heat equation leading to a non-standard coupled governing problem. The goal of this work is to find solution estimate since the exact solution of the studied problem is not possible to determine. We use an asymptotic expansion in order of a small parameter which is presented as a heat expansion coefficient of the pipe's material. Furthermore, an error estimate is provided for the mentioned asymptotic approximation of the solution for inner area of the pipe. Close to the boundary, problem becomes more complex so different approaches are observed, mainly Theory of Perturbations and Separations of Variables. In view of that, error estimate for the whole approximation will be provided with additional software simulations of gotten situation.
Fuzzy Logic and Control Strategies on a Sump
Sump can be defined as a reservoir which contains slurry; a mixture of solid and liquid or water, in it. Sump system is an unsteady process owing to the level response. Sump level shall be monitored carefully by using a good controller to avoid overflow. As the current conventional controllers unable to solve problems with time delay and nonlinearities, Fuzzy Logic controller is tested to prove its ability in solving the listed problems of slurry sump. Therefore, in order to justify the effectiveness and reliability of these controllers, simulation of the sump system was created by using MATLAB and the results were compared. According to the result obtained, instead of Proprotional-Integral (PI) and Proportional-Integral and Derivative (PID), Fuzzy Logic controller showed the best result by offering quick response of 0.32s for step input and 5s for pulse generator, to the set point change as well as producing the minimal Integral Absolute Error (IAE) value that is 0.66 and 0.36.
Topological Language for Classifying Linear Chord Diagrams via Intersection Graphs
Chord diagrams occur in mathematics, from the study of RNA to knot theory. They are widely used in theory of knots and links for studying the finite type invariants, whereas in molecular biology one important motivation to study chord diagrams is to deal with the problem of RNA structure prediction. An RNA molecule is a linear polymer, referred to as the backbone, that consists of four types of nucleotides. Each nucleotide is represented by a point, whereas each chord of the diagram stands for one interaction for Watson-Crick base pairs between two nonconsecutive nucleotides. A chord diagram is an oriented circle with a set of n pairs of distinct points, considered up to orientation preserving diffeomorphisms of the circle. A linear chord diagram (LCD) is a special kind of graph obtained cutting the oriented circle of a chord diagram. It consists of a line segment, called its backbone, to which are attached a number of chords with distinct endpoints. There is a natural fattening on any linear chord diagram; the backbone lies on the real axis, while all the chords are in the upper half-plane. Each linear chord diagram has a natural genus of its associated surface. To each chord diagram and linear chord diagram, it is possible to associate the intersection graph. It consists of a graph whose vertices correspond to the chords of the diagram, whereas the chord intersections are represented by a connection between the vertices. Such intersection graph carries a lot of information about the diagram. Our goal is to define an LCD equivalence class in terms of identity of intersection graphs, from which many chord diagram invariants depend. For studying these invariants, we introduce a new representation of Linear Chord Diagrams based on a set of appropriate topological operators that permits to model LCD in terms of the relations among chords. Such set is composed of: crossing, nesting, and concatenations. The crossing operator is able to generate the whole space of linear chord diagrams, and a multiple context free grammar able to uniquely generate each LDC starting from a linear chord diagram adding a chord for each production of the grammar is defined. In other words, it allows to associate a unique algebraic term to each linear chord diagram, while the remaining operators allow to rewrite the term throughout a set of appropriate rewriting rules. Such rules define an LCD equivalence class in terms of the identity of intersection graphs. Starting from a modelled RNA molecule and the linear chord, some authors proposed a topological classification and folding. Our LCD equivalence class could contribute to the RNA folding problem leading to the definition of an algorithm that calculates the free energy of the molecule more accurately respect to the existing ones. Such LCD equivalence class could be useful to obtain a more accurate estimate of link between the crossing number and the topological genus and to study the relation among other invariants.
Study and Analysis of a Susceptible Infective Susceptible Mathematical Model with Density Dependent Migration
In this paper, a susceptible infective susceptible mathematical model is proposed and analyzed where the migration of human population is given by migration function. It is assumed that the disease is transmitted by direct contact of susceptible and infective populations with constant contact rate. The equilibria and their stability are studied by using the stability theory of ordinary differential equations and computer simulation. The model analysis shows that the spread of infectious disease increases when human population immigration increases in the habitat but it decreases if emigration increases.
Singular Perturbed Vector Field Method Applied to the Problem of Thermal Explosion of Polydisperse Fuel Spray
In our research, we present the concept of singularly perturbed vector field (SPVF) method, and its application to thermal explosion of diesel spray combustion. Given a system of governing equations, which consist of hidden Multi-scale variables, the SPVF method transfer and decompose such system to fast and slow singularly perturbed subsystems (SPS). The SPVF method enables us to understand the complex system, and simplify the calculations. Later powerful analytical, numerical and asymptotic methods (e.g method of integral (invariant) manifold (MIM), the homotopy analysis method (HAM) etc.) can be applied to each subsystem. We compare the results obtained by the methods of integral invariant manifold and SPVF apply to spray droplets combustion model. The research deals with the development of an innovative method for extracting fast and slow variables in physical mathematical models. The method that we developed called singular perturbed vector field. This method based on a numerical algorithm applied to global quasi linearization applied to given physical model. The SPVF method applied successfully to combustion processes. Our results were compared to experimentally results. The SPVF is a general numerical and asymptotical method that reveals the hierarchy (multi-scale system) of a given system.
A Study of Non Linear Partial Differential Equation with Random Initial Condition
In this work, we present the effect of noise on the solution of a partial differential equation (PDE) in three different setting. We shall first consider random initial condition for two nonlinear dispersive PDE the non linear Schrodinger equation and the Kortteweg –de vries equation and analyse their effect on some special solution , the soliton solutions.The second case considered a linear partial differential equation , the wave equation with random initial conditions allow to substantially decrease the computational and data storage costs of an algorithm to solve the inverse problem based on the boundary measurements of the solution of this equation. Finally, the third example considered is that of the linear transport equation with a singular drift term, when we shall show that the addition of a multiplicative noise term forbids the blow up of solutions under a very weak hypothesis for which we have finite time blow up of a solution in the deterministic case. Here we consider the problem of wave propagation, which is modelled by a nonlinear dispersive equation with noisy initial condition .As observed noise can also be introduced directly in the equations.
On Chvátal's Conjecture for the Hamiltonicity of 1-Tough Graphs and Their Complements
Graph toughness and the associated cycle structure have attracted much attention and aroused extensive works since Chvátal introduced this concept in 1973. Among the seven conjectures posted by then, much fewer results were published for the one relating the existence of a hamiltonian cycle in any 1-tough graph to its complement graph. In this paper, we show that the conjecture does not hold in general. More precisely, it is true only for graphs with six or seven vertices and is false for graphs with eight or more vertices. A new theorem is derived as a correction for the conjecture.
One-Step Time Series Predictions with Recurrent Neural Networks
Time series prediction problems have many important practical applications, but are notoriously difficult for statistical modeling. Recently, machine learning methods have been attracted significant interest as a practical tool applied to a variety of problems, even though developments in this field tend to be semi-empirical. This paper explores application of Long Short Term Memory based Recurrent Neural Networks to the one-step prediction of time series for both trend and stochastic components. Two types of data are analyzed - daily stock prices, that are often considered to be a typical example of a random walk, - and weather patterns dominated by seasonal variations. Results from both analyses are compared, and reinforced learning framework is used to select more efficient between Recurrent Neural Networks and more traditional auto regression methods. It is shown that both methods are able to follow long-term trends and seasonal variations closely, but have difficulties with reproducing day-to-day variability. Future research directions and potential real world applications are briefly discussed.
Science, Technology, Engineering And Math Approach on Playing Video Games
This STEM project is to demonstrate how to play Video Game through STEM approach and learning science and statistics. Kids are playing video games too long and parents do not want kids to play video games since most video games are not developing critical thinking. Hill Climb Car Racing game was chosen not based on commercial rating but on potential of applying statistical data-driven methodology. The author took STEM approach: Science (Physics, Mechanics), Technology (Car Upgrading), Engineering (Failure Mode), and Math (Geometry, Trigonometry and Statistics). Based on the engineering failure mode analysis and scientific understanding, author can develop a systematic car upgrading system through statistical modeling to optimize car performance. Simple linear regression was conducted to quantify the Performance Index (ROI) return (car travelling distance) of investment (car grading cost). The regression model accuracy has been improved from original 66% (random playing mode) to 92% (systematic playing mode). The ROI slope has been improved from 147.2 to 512.4 meter/upgrade unit. The clustering analysis grouped the similar field stages with common challenges and science which has helped upgrade car to support multiple stages. The best car of each stage has matched well with literature research. It’s a very successful STEM project on playing video games.
Boundary Condition with the Riemann-Liouville Fractional Time Derivative at a Thin Membrane for Normal Diffusion
In many physical models concerning various diffusion processes, fractional derivatives are involved in diffusion equations. However, in many cases the interpretation of the equations is vague. In practice, an experimental verification of fractional models is also a problem. In our contribution, we study normal diffusion in a system with a thin membrane. We show the method of deriving boundary conditions at the membrane directly from experimental data. One of the conditions contains the Riemann-Liouville time fractional derivative of 1/2 order. Such boundary condition is rather unexpected since, as far as we know, there is no need to involve fractional time derivative in normal diffusion model. We analyze an influence of fractional derivative occurring in the boundary condition on diffusion process. The physical interpretation of the boundary condition is given as well.
The Effects of Mobile Communication on the Nigerian Populace
Communication, the activity of conveying information, remains a vital resource for the growth and development of any given society. Mobile communication, popularly known as global system for mobile communication (GSM) is a globally accepted standard for digital cellular communication. GSM, which is a wireless technology, remains the fastest growing communication means worldwide. Indeed, mobile phones have become a critical business tool and part of everyday life in both developed and developing countries. This study examines the effects of mobile communication on the Nigerian populace. The methodology used in this study is the survey research method with the main data collection tool as questionnaires. The questionnaires were administered to a total of seventy respondents in five cities across the country, namely: Aba, Enugu, Bauchi, Makurdi, and Lagos. The result reveals that though there is some quality of service issues, mobile communication has very significant positive efforts on the economic and social development of the Nigerian populace.
Hmamiltonian Paths and Cycles Passing through Prescribed Edges in the Balanced Hypercubes
The n-dimensional balanced hypercube BHn (n ≥ 1) has been proved to be a bipartite graph. Let P be a set of edges whose induced subgraph consists of pairwise vertex-disjoint paths. For any two vertices u, v from different partite sets of V (BHn). In this paper, we prove that if |P| ≤ 2n − 2 and the subgraph induced by P has neither u nor v as internal vertices, or both of u and v as end-vertices, then BHn contains a Hamiltonian path joining u and v passing through P. As a corollary, if |P| ≤ 2n−1, then the BHn contains a Hamiltonian cycle passing through P.
Multiscale Simulation of Absolute Permeability in Carbonate Samples Using 3D X-Ray Micro Computed Tomography Images Textures
Characterizing rock properties of carbonate reservoirs is highly challenging because of rock heterogeneities revealed at several length scales. In the last two decades, the Digital Rock Physics (DRP) approach was implemented successfully in sandstone rocks reservoirs in order to understand rock properties behaviour at the pore scale. This approach uses 3D X-ray Microtomography images to characterize pore network and also simulate rock properties from these images. Even though, DRP is able to predict realistic rock properties results in sandstone reservoirs it is still suffering from a lack of clear workflow in carbonate rocks. The main challenge is the integration of properties simulated at different scales in order to obtain the effective rock property of core plugs. In this paper, we propose several approaches to characterize absolute permeability in some carbonate core plugs samples using multi-scale numerical simulation workflow. In this study, we propose a procedure to simulate porosity and absolute permeability of a carbonate rock sample using textures of Micro-Computed Tomography images. First, we discretize X-Ray Micro-CT image into a regular grid. Then, we use a textural parametric model to classify each cell of the grid using supervised classification. The main parameters are first and second order statistics such as mean, variance, range and autocorrelations computed from sub-bands obtained after wavelet decomposition. Furthermore, we fill permeability property in each cell using two strategies based on numerical simulation values obtained locally on subsets. Finally, we simulate numerically the effective permeability using Darcy’s law simulator. Results obtained for studied carbonate sample shows good agreement with the experimental property.
Bulk Viscous Bianchi Type V Cosmological Model with Time Dependent Gravitational Constant and Cosmological Constant in General Relativity
In this paper, we investigate Bulk Viscous Bianchi Type V Cosmological Model with Time dependent gravitational constant and cosmological constant in general Relativity by assuming ξ(t)=ξ_(0 ) p^m where ξ_(0 ) and m are constants. We also assume a variation law for Hubble parameter as H(R) = a (R^(-n)+1), where a>0, n>1 being constant. Two universe models were obtained, and their physical behavior has been discussed. When n=1 the Universe starts from singular state whereas when n=0 the cosmology follows a no singular state. The presence of bulk viscosity increase matter density’s value.
Convergence Analysis of Cubic B-Spline Collocation Method for Time Dependent Parabolic Advection-Diffusion Equations
A comprehensive numerical study is presented for the solution of time-dependent advection diffusion problems by using cubic B-spline collocation method. The linear combination of cubic B-spline basis, taken as approximating function, is evaluated using the zeros of shifted Chebyshev polynomials as collocation points in each element to obtain the best approximation. A comparison, on the basis of efficiency and accuracy, with the previous techniques is made which confirms the superiority of the proposed method. An asymptotic convergence analysis of technique is also discussed, and the method is found to be of order two. The theoretical analysis is supported with suitable examples to show second order convergence of technique. Different numerical examples are simulated using MATLAB in which the 3-D graphical presentation has taken at different time steps as well as different domain of interest.
Mathematical Modeling and Analysis of Forced Vibrations in Micro-Scale Microstretch Thermoelastic Simply Supported Beam
The present paper deals with the flexural vibrations of homogeneous, isotropic, generalized micropolar microstretch thermoelastic thin Euler- Bernoulli beam resonators, due to exponential time varying load. Both the axial ends of the beam are assumed to be at simply supported conditions. The governing equations have been solved analytically by using Laplace transforms technique twice with respect to time and space variables respectively. The inversion of Laplace transform in time domain has been performed by using the calculus of residues to obtain deflection. The analytical results have been numerically analyzed with the help of MATLAB software for magnesium like material. The graphical representations and interpretations have been discussed for deflection of beam under simply supported boundary condition and for distinct considered values of time and space as well. The obtained results are easy to implement for engineering analysis and designs of resonators (sensors), modulators, actuators.
Asymptotic Expansion of the Korteweg-de Vries-Burgers Equation
It is common knowledge that many physical problems (such as non-linear shallow-water waves and wave motion in plasmas) can be described by the Korteweg-de Vries (KdV) equation, which possesses certain special solutions, known as solitary waves or solitons. As a marriage of the KdV equation and the classical Burgers (KdVB) equation, the Korteweg-de Vries-Burgers (KdVB) equation is a mathematical model of waves on shallow water surfaces in the presence of viscous dissipation. Asymptotic analysis is a method of describing limiting behavior and is a key tool for exploring the differential equations which arise in the mathematical modeling of real-world phenomena. By using variable transformations, the asymptotic expansion of the KdVB equation is presented in this paper. The asymptotic expansion may provide a good gauge on the validation of the corresponding numerical scheme.
A Simple Finite Element Method for Glioma Tumor Growth Model with Density Dependent Diffusion
In this presentation, we have performed numerical simulations for a reaction-diffusion equation with various nonlinear density-dependent diffusion operators and proliferation functions. The mathematical model represented by parabolic partial differential equation is considered to study the invasion of gliomas (the most common type of brain tumors) and to describe the growth of cancer cells and response to their treatment. The unknown quantity of the given reaction-diffusion equation is the density of cancer cells and the mathematical model based on the proliferation and migration of glioma cells. A standard Galerkin finite element method is used to perform the numerical simulations of the given model. Finally, important observations on the each of nonlinear diffusion functions and proliferation functions are presented with the help of computational results.
Mathematical Analysis of Simple Supported Euler-Bernoulli Beam on a Variable Elastic Foundation under a Partially Distributed Moving Load
The dynamic responses of an elastically supported Euler- Bernoulli beam on variable elastic foundation under partially distributed moving loads were investigated. The governing equation is fourth order partial differential equation, which was reduced to second order ordinary differential equation by using the analytical method in terms of series solution and solved by a numerical method using mathematical software (Maple). The numerical analysis shows that the response amplitude of the moving mass and moving force for variable pre-stressed increase as mass of the load M increases. It was found that the response displacement of the beam decreases as the value of the elastic foundation K increases. Also, the response displacement of the beam decreases as the value of the pre-stressed N increase. Comparison of moving mass and moving force shown that moving mass is greater than that of moving force.
Statistical Convergence for the Approximation of Linear Positive Operators
In this paper, we consider positive linear operators and study the Voronovskaya type result of the operator then obtain an error estimate in terms of the higher order modulus of continuity of the function being approximated and its A-statistical convergence. Also, we compute the corresponding rate of A-statistical convergence for the linear positive operators.