Ordinary Differential Equations (ODE)

Nipah virus (NiV) is a newly detected highly pathogenic virus with ability to cause devastating morbidity and mortality (an estimated 100% in some cases) rate among the human populations. This emerging infectious disease has become one of the most alarming threats of the public healths in Bangladesh mainly due to its periodic outbreaks (as it strikes almost every year) and the highly devastating mortality rate. In this paper, we propose a mathematical model describing the host-pathogen interactions in terms of ordinary differential equations (ODEs). The main aim is to investigate the disease propagation and control strategy of NiV infections. The behaviour of the dynamics of NiV infections has been illustrated by the numerical simulations.

For $p\geq 2$ , let $E$ be a 2-uniformly smooth and $p$ -uniformly convex real Banach space and let $A:E\rightarrow E^{\ast }$ be a Lipschitz and strongly monotone mapping such that $A^{-1}(0)\neq \emptyset$ . For given $x_{1}\in E$ , let $\{x_{n}\}$ be generated by the algorithm $x_{n+1}=J^{-1}(Jx_{n}-\unicode[STIX]{x1D706}Ax_{n})$ , $n\geq 1$ , where $J$ is the normalized duality mapping from $E$ into $E^{\ast }$ and $\unicode[STIX]{x1D706}$ is a positive real number in $(0,1)$ satisfying suitable conditions. Then it is proved that $\{x_{n}\}$ converges strongly to the unique point $x^{\ast }\in A^{-1}(0)$ . Furthermore, our theorems provide an affirmative answer to the Chidume et al. open problem [‘Krasnoselskii-type algorithm for zeros of strongly monotone Lipschitz maps in classical Banach spaces’, SpringerPlus4 (2015), 297]. Finally, applications to convex minimization problems are given.

Article history: Received 04 March 2019 Revised 10 March 2019 Accepted 20 March 2019 Available online 5 April 2019 In this work, deterministic model has been used to construct a transmission dynamics of measles using five compartmental models namely; MSEIR. The Disease Free Equilibrium (DFE) point effective reproduction number and the basic reproduction number for the model were obtained. The simulations of the ordinary differential equation (ODEs) and the reproduction number were established. Simulations of different variables of the model have been performed. The sensitivity analysis of different embedded parameters revealed that the proportion of the immunized population exceeded the herd immunity level of measles. Therefore, the disease cannot persist in the population.

ABSTRACT Differential equations play a noticeable role in engineering, physics, economics, and other disciplines. In this paper, a general approach is suggested to solve a wide variety of linear and nonlinear ordinary differential equations (ODEs) that are independent of their forms, orders, and given conditions. With the aid of certain fundamental concepts of mathematics, Fourier series expansion and metaheuristic methods, ODEs can be represented as an optimization problem. The target is to minimize the weighted residual function (cost function) of the ODEs. To this end, two different approaches, unit weight function and least square weight function, are examined in order to determine the appropriate method. The boundary and initial values of ODEs are considered as constraints for the optimization model. Generational distance metric is used for evaluation and assessment of the approximate solutions versus the exact solutions. Six ODEs and four mechanical problems are approximately solved and compared with their exact solutions. The optimization task is carried out using different optimizers including the particle swarm optimization, the cuckoo search, and the water cycle algorithm. The optimization results obtained show that metaheuristic algorithms can be successfully applied for approximate solving of different types of ODEs. The suggested least square weight function is slightly superior over the unit weight function in terms of accuracy and statistical results for approximate solving of ODEs.

Being able to transform an analog audio circuit into a digital model is a big deal for musicians, producers, and circuit benders alike. In this paper, we address some of the issues that arise when attempting to make such a digital model. Using the canonical state variable filter as the main point of interest in our schematic, we will walk through the process of making a signal flow graph, obtaining a transfer function, and making a usable digital filter. Additionally, we will address an issue that is common throughout virtual analog literature; reducing the very large expressions for each of the filter coefficients. Using a novel factoring algorithm, we show that these expressions can be reduced from thousands of operations down to tens of operations.

Abstract: In this paper, Variation of Parameters Method (VPM) is used to find the analytical solutions of non-linear fractional order quadratic Riccati differential equation. The given method is applied to initial value problems of the fractional order Riccati differential equations. The proposed technique has no discretization, linearization, perturbation, transformation, preventive suspicions and it is also free from Adomian,s polynomials. The obtained results are compare with analytical solutions by graphical and tabular form.

... All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Page 3. BRIEF TABLE OF INTEGRALS 1. 3. 5. 7. 9. 11. 13. 15. 17. 19. 21. 23. 25. 27. 29. 31. 33. 35. 37. 39. 41. 43. -+C, n -1 u" du = n + 1 e"du=e" +C sin udu = -cos u+C sec2 udu = tan u + C sec ...

Elementary Differential Equations with Boundary Value ProblemsStudent Solutions Manual, Partial Differential Equations & Boundary Value Problems with MapleElementary Differential Equations with Boundary Value ProblemsPartial Differential Equations and Boundary-value Problems with ApplicationsDifferential Equations with Boundary Value ProblemsElementary Differential EquationsElementary Differential Equations with Boundary Value Problems,Elementary Differential Equations and Boundary Value Problems, Solutions ManualIntroduction to Differential Equations with Boundary Value ProblemsPartial Differential Equations and Boundary Value ProblemsBoundary Value Problems from Higher Order Differential EquationsApplied Differential Equations with Boundary Value ProblemsDifferential Equations with Boundary Value ProblemsOrdinary Differential Equations And Boundary Value Problems Volume Ii: Boundary Value ProblemsDifferential Equations with Boundary-value ProblemsStudent Solutions Manual for Zill/Wright's Differential Equations with Boundary-Value Problems, 8thDifferential Equations and Boundary Value ProblemsElementary Partial Differential Equations with Boundary Value ProblemsFundamentals of Differential EquationsElementary Differential Equations and Boundary Value ProblemsDifferential Equations and Boundary Value ProblemsCodes for Boundary-Value Problems in Ordinary Differential EquationsPartial Differential Equations and Boundary Value ProblemsDifferential Equations with Boundary-Value ProblemsPartial Differential Equations with Fourier Series and Boundary Value ProblemsDifferential Equations and Boundary Value ProblemsElementary Differential Equations and Boundary Value Problems, 11e Student Solutions ManualPartial Differential Equations with Fourier Series and Boundary Value ProblemsA First Course in Differential Equations with Modeling ApplicationsDifferential Equations and Boundary Value ProblemsBoundary Value Problems for Systems of Differential, Difference and Fractional EquationsDifferential Equations with Boundary Value ProblemsFundamentals of Differential Equations and Boundary Value ProblemsBoundary Value ProblemsElementary Differential Equations and Boundary Value ProblemsA Course in Differential Equations with Boundary Value ProblemsElementary Differential Equations and Boundary Value ProblemsDifferential Equations with Boundary-Value ProblemsOrdinary Differential Equations and Boundary Value ProblemsDifferential Equations and Boundary Value Problems

In this manuscript the implicit Runge-Kutta (IRK) method, with three slopes of order five has been explained, and is applied to Van der pol stiff differential equation. Truncation error, of order five, has been estimated.  Stability of the procedure for the Van der pol equation, is analyzed by the Lyapunov method. To illustrate the structure of the method, an Algorithm is presented to solve this stiff problem. Results confirm the validity and the ability of this approach.

We present some implicit methods to approximate the zeros of monotone operators in the setting of Banach spaces. The methods considered herein converge strongly to the desired solutions under certain assumptions. As applications, we employ our methods to obtain solutions of convex minimization problems and Fredholm integral equations. Finally, we show the effectiveness and efficiency of the algorithm considered herein.

After more than 30 years of continuous research as well as unselfish efforts, tremendous and exciting developments have been achieved towards the evolution of HIV treatments both in the directions of antiretroviral therapy and effective vaccine for HIV positive patients. Recent research shows that triple-drug antiretroviral therapy can 'functionally cure' [1, 2] the HIV positive patients, which is a milestone in the therapeutic treatments of AIDS. Despite the significant progress on the evolution of AIDS/HIV treatments, it is still a curse for the humanity and until today the world's most serious epidemic as well as leading infectious killer disease. However, Millennium Development Goals (MDGs) Getting to Zero shows a hope to build up a new world without AIDS. In this paper we investigate and focus on those issues of the overall present scenarios of AIDS treatment for the preventive strategies of HIV infections. A mathematical model is analyzed with numerical simulations showing its importance in diseases control and preventions. Optimal control theory is applied to compare the results in the presence of state constraints.

Using mathematical methods to understand and model crime is a recent idea that has drawn considerable attention from researchers during the last five years. From the plethora of models that have been proposed, perhaps the most successful one has been a diffusion-type differential equations model that describes how the number of criminals evolves in a specific area. We propose a more detailed form of this model that allows for two distinct criminal types associated with major and minor crime. Additionally, we examine a stochastic variant of the model that represents more realistically the ‘generation’ of new criminals. Numerical solutions from both models are presented and compared with actual crime data for the Greater Manchester area. Agreement between simulations and actual data is satisfactory. A preliminary statistical analysis of the data also supports the model's potential to describe crime.

Mathematics 'Purest of the Pure Science' traditionally known as 'Dirty and Dust' is actually a subject of 'Beauty and Trust'. On one hand it is the backbone of the field of knowledge, and on the other hand each face of the universe is governed by some mathematical rule. Mathematics on which various disciplines of science and technology are found to rest is continuously extending its domain of influence on other branches of knowledge. Present note is a little effort to throw some light on different applications of mathematics, for the betterment of society, with special reference to its wide applications in fashion technology, an emerging subject of the field of knowledge.

The information has been analytically concentrated for an aggravation mode by reduction of modelled relations to dominant positive feedback loops; the doomsday of capitalist reproduction because of neglect of the scientific foundations of environmental management is determined by initial magnitudes of only two variables - the resource intensity of net output and the ratio of net output to proved reserves in the model economy. The magnitudes of remaining parameters (relating, first, to technical progress, distribution shares of labour compensation, rent, and profit in the net product, second, to the rate of growth of labor force and the rate of investment in fixed capital) affect only the nuances of a transition regime to a collapse.

In this paper, the four-stage fourth-order Runge-Kutta-Nyström method (RKN) is used for solving second-order two-point boundary value problems (BVP) with Dirichlet condition. The method obtained the solution of the second-order boundary value problem directly without reducing it to first-order equations. The method is implemented using a constant step size via shooting technique. Numerical results are compared with the existing Runge-Kutta (RK) method and have clearly shown the advantage and the efficiency of the RKN.

Enzymatic reactions can be simulated by means of non-linear ODEs (ordinary differential equations) of the first order. Hence, the study of the asymptotic behavior of the solutions of this kind of ODEs is relevant for the comprehension and simulation of cellular metabolism.

In this paper, Variation of Parameters Method (VPM) is used to find the analytical solutions of non-linear fractional order quadratic Riccati differential equation. The given method is applied to initial value problems of the fractional order Riccati differential equations. The proposed technique has no discretization, linearization, perturbation, transformation, preventive suspicions and it is also free from Adomian,s polynomials. The obtained results are compare with analytical solutions by graphical and tabular form.

Calculation of eigenvalue problems for the second-order ordinary differential equations is relevant for the physics problems. The second-order ordinary differential equation with homogeneous Dirichlet boundary condition was considered. The Chebyshev pseudospectral method (CPM) was used for the problem of eigenvalues basing on the Chebyshev-Gauss-Lobatto points to create the differential matrices. The Mathematica version 10.4 to write computing programs was used. In the applications, the Chebyshev pseudospectral method was used to find eigenvalues that were approximated gradually to the exact eigenvalues of the problem.

reproduction number and the basic reproduction number for the model Available online 5 April 2019 were obtained. The simulations of the ordinary differential equation (ODEs) and the reproduction number were established. Simulations of different variables of the model have been performed. The sensitivity analysis of different embedded parameters revealed that the proportion of the immunized population exceeded the herd immunity level of measles. Therefore, the disease cannot persist in the population.

Linear and nonlinear wave equations are important in various aspects of
mathematics and physics, including the study of black holes. After a brief
survey of the sine-Gordon equation (SGE), this report presents a fourth-order
numerical approximation scheme for solving equations of this type. This is
followed by a numerical investigation of SGE in the presence of the Poschl-
Teller potential. We close with a discussion of the application of this method
to the study of black hole dynamics.

In this paper, numerical solutions of the Newell Whitehead Segel
Type equations (NWS) are obtained using di�erent cubic B-spline basis. A
linear Von-Neumann stability analysis shows that the numerical scheme is
unconditionally stable. Accuracy of the method is discussed by computing the
max numerical error. The numerical result shows that the presented method
is a successful numerical technique for solving NWS equation.

Nipah virus (NiV) is a newly detected highly pathogenic virus with ability to cause devastating morbidity and mortality (an estimated 100% in some cases) rate among the human populations. This emerging infectious disease has become one of the most alarming threats of the public healths in Bangladesh mainly due to its periodic outbreaks (as it strikes almost every year) and the highly devastating mortality rate. In this paper, we propose a mathematical model describing the host-pathogen interactions in terms of ordinary differential equations (ODEs). The main aim is to investigate the disease propagation and control strategy of NiV infections. The behaviour of the dynamics of NiV infections has been illustrated by the numerical simulations.

This study will explicitly demonstrate by example that an unrestricted infinite and forward recursive hierarchy of differential equations must be identified as an unclosed system of equations, despite the fact that to each unknown function in the hierarchy there exists a corresponding determined equation to which it can be bijectively mapped to. As a direct consequence, its admitted set of symmetry transformations must be identified as a weaker set of indeterminate equivalence transformations. The reason is that no unique general solution can be constructed, not even in principle. Instead, infinitely many disjoint and thus independent general solution manifolds exist. This is in clear contrast to a closed system of differential equations that only allows for a single and thus unique general solution manifold, which, by definition, covers all possible particular solutions this system can admit. Herein, different first order Riccati-ODEs serve as an example, but this analysis is not restricted to them. All conclusions drawn in this study will translate to any first order or higher order ODEs as well as to any PDEs.

In this paper, Variation of Parameters Method (VPM) is used to find the analytical solutions of non-linear fractional order quadratic Riccati differential equation. The given method is applied to initial value problems of the fractional order Riccati differential equations. The proposed technique has no discretization, linearization, perturbation, transformation, preventive suspicions and it is also free from Adomian's polynomials. The obtained results are compare with analytical solutions by graphical and tabular form.

We present a novel framework for developing Wave Digital Filter (WDF) models from reference circuits with multiple/multiport nonlinearities. Collecting all nonlinearities into a vector at the root of a WDF tree bypasses the traditional WDF limitation to a single nonlinearity. The resulting system has a complicated scattering relationship between the nonlinearity ports and the ports of the rest of the (linear) circuit, which can be solved by a Modified-NodalAnalysis-derived method. For computability reasons, the scattering and vector nonlinearity must be solved jointly; we suggest a derivative of the K-method. This novel framework significantly expands the class of appropriate WDF reference circuits. A case study on a clipping stage from the Big Muff Pi distortion pedal involves both a transistor and a diode pair. Since it is intractable with standard WDF methods, its successful simulation demonstrates the usefulness of the novel framework.

This paper somewhat humorously shows a physical model of the semi-fantastic idea of an underground tunnel that would connect the two cities, which would be traveled through by the so called gravity train. The idea has existed since Robert Hook first presented it to Sir Isaac Newton. This problem is described by the system of ordinary differential equations. The solution of equation system is obtained numerically using Mathcad applications (with units) and function Odesolve.

In this paper, numerical solutions of the Newell Whitehead Segel Type equations (NWS) are obtained using di�erent cubic B-spline basis. A linear Von-Neumann stability analysis shows that the numerical scheme is unconditionally stable. Accuracy of the method is discussed by computing the max numerical error. The numerical result shows that the presented method is a successful numerical technique for solving NWS equation.

The objective of this course is to acquaint the students with differential equations and their applications in engineering. Course contents: Differential equations and their classification, formation of differential equations, solution of differential equations, initial and boundary conditions, methods of solution of differential equation of first order and first-degree: separable equations, homogeneous equations, equations reducible to homogeneous, exact differential equations, integrating factor, linear equations, Bernoulli equations, orthogonal trajectories in cartesian and polar coordinates, applications of first order differential equations. Nonlinear first order differential equations. Non-linear first order differential equations: Equations solvable for p, for y and for x, Clairaut's equations. Homogeneous linear equations of order n with constant coefficients, auxiliary/characteristics equations. Solution of higher order differential equation according to the roots of auxiliary equation. (Real and distinct, real and repeated, and complex). Non-homogeneous linear equations. Working rules for finding particular integral. Cauchy Euler equation. Applications of higher order linear differential equations.

We present a novel framework for developing Wave Digital Filter (WDF) models from reference circuits with multiple/multiport nonlinearities. Collecting all nonlinearities into a vector at the root of a WDF tree bypasses the traditional WDF limitation to a single nonlinearity. The resulting system has a complicated scattering relationship between the nonlinearity ports and the ports of the rest of the (linear) circuit, which can be solved by a Modified-Nodal- Analysis-derived method. For computability reasons, the scattering and vector nonlinearity must be solved jointly; we suggest a derivative of the K-method. This novel framework significantly expands the class of appropriate WDF reference circuits. A case study on a clipping stage from the Big Muff Pi distortion pedal involves both a transistor and a diode pair. Since it is intractable with standard WDF methods, its successful simulation demonstrates the usefulness of the novel framework.

We present a Modified-Nodal-Analysis-derived method for developing Wave Digital Filter (WDF) adaptors corresponding to complicated (non-series/parallel) topologies that may include multi-port linear elements (e.g. controlled sources and transformers). A second method resolves noncomputable (non-tree-like) arrangements of series/parallel adaptors. As with the familiar 3-port series and parallel adaptors, one port of each derived adaptor may be rendered reflection-free, making it acceptable for inclusion in a standard WDF tree. With these techniques, the class of acceptable reference circuits for WDF modeling is greatly expanded. This is demonstrated by case studies on circuits which were previously intractable with WDF methods: the Bassman tone stack and Tube Screamer tone/volume stage.

The starting point for this paper is a nonlinear, two-point boundary value ordinary differential equation (BVODE) that defines corneal curvature according to a static force balance. A numerical solution to the BVODE is computed by first converting the BVODE to a parabolic partial differential equation (PDE) by adding an initial value (t, pseudo-time) derivative to the BVODE. A numerical solution to the PDE is then computed by the method of lines (MOL) with the calculation proceeding to a sufficiently large value of t such that the derivative in t reduces to essentially zero. The PDE solution at this point is also the solution for the BVODE. This procedure is implemented in R (an open source scientific programming system) and the programming is discussed in some detail. A series approximation to the solution is derived from which an estimate for the rate of convergence is obtained. This is compared to a fitted exponential model. Also, two linear approximations are derived, one of whi...

In this study, an exact Eulerian solution is derived for the bidirectional vortex in a conical chamber. Our model is applicable to idealized representations of cyclone separators and liquid rocket engines with slowly expanding chamber cross-sections. The bulk fluid motion is assumed to be nonreactive, steady, rotational, inviscid, and incompressible. Our approach is founded on the Bragg-Hawthorne equation and seeks to overcome some of the deficiencies encountered by Bloor and Ingham (Bloor, M. I. G., and Ingham, D. B., “The Flow in Industrial Cyclones,” Journal of Fluid Mechanics, Vol. 178, 1987, pp. 507-519). Despite inevitable similarities with Bloor and Ingham’s model, ours is constructed using a judicious framework that connects the solution to the swirl number and the cone divergence angle. In consequence, a self-similar formulation is produced that is independent of the cone’s finite body length. closed-form representations for the principal variables This enables us able to c...