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This paper studies the propagation of monetary shocks in an economy featuring a strategic microfoundation for price rigidities. Following an aggregate shock to money, most consumers are initially uninformed. The market for goods is decentralized. Firms are better off delaying the adjustment of prices until enough consumers learn. At the same time, consumers learn from firms that have adjusted prices. The implied endogenous information diffusion follows a Bernoulli differential equation, implying a nonlinear path of learning. Nonlinear learning implies hump-shaped dynamics of output and inflation. A quantitative exercise suggests that these dynamics can be sizable and persistent. (JEL D11, D21, D40, D82, E23, E31)

In this paper, we prove that a class of differential system of degree nine is Liouville integrable by transforming it into a Bernoulli differential equation and we determine exactly its first integral. This allows us to show that this class admits an explicit non-algebraic limit cycle enclosing the origin, here a non-elementary singular point. For singularities, at infinity, this class does not possess singular points.

We present the exact analytic solution to the nonlinear dynamic system describing the information propagation process in complex networks. The method is to switch to the dynamic system of equations for the new variables defined according to the nonlinear terms appearing in the original dynamic system, and then, after combining relevant consistent condition of solution to reduce the dynamic system of equations for the newly defined variables to the knownly solvable nonlinear Bernoulli differential equation. Numerical comparisons between the purely numerical solution and the exact analytic solution confirm the crosscheck and mutual proof for both of the solutions. The presented exact analytic solution, which does not restrict the coefficients in the original dynamic system to merely constants, is thus suitable for the study of information propagation in networks with evolving structure and changing properties.

The equilibrium shapes of lipid vesicles are governed by the general shape equation which is derived from the minimization of the Helfrich free energy and can be reduced to the Willmore equation in a special case. The general shape equation is a high-order nonlinear partial differential equation and it is very difficult to find analytical solution even in axisymmetric case, which is reduced to a second-order ordinary differential equation. In the traditional axisymmetric shape equation, the turning radius is the variable. Here we study the shape equation by choosing the tangential angle as the variable. In this case, the Willmore equation is reduced to the Bernoulli differential equation and the general solution is obtained conveniently. We find that the curvature in this solution is discontinuous in some cases, which was not noticed previously. This solution can satisfy the boundary conditions for an open vesicle with free edges.

In this paper a numerical method is given for the solution of linear Fredholm integro differential equation with piecewise intervals under the mixed conditions using the Bernoulli polynomials. The aim of this article is to present an efficient numerical procedure for solving linear Fredholm integro-differential equation with piecewise intervals. This method transforms linear Fredholm integro-differential equation with piecewise intervals and the given conditions into matrix equation which corresponds to a system of linear algebraic equation. Finally, some experiments and their numerical solutions are given. The results reveal that this method is reliable and efficient.

We consider solutions to the semilinear ordinary differential equation with a nonlinear term of Emden–Fowler type. The results about the asymptotic behavior of the solutions to the Emden–Fowler equation defined in a neighborhood of infinity, presented in the book of R. Bellman, are extended to the case of equation with lower-order derivative.

In this paper, appropriate Lie group transformations are considered as an alternative to a travelling wave transformation, in the auxiliary equation method to solve the nonlinear partial differential equations. In addition, Bernoulli differential equation has been considered as an auxiliary equation and using the approach presented.

This paper concerns a probabilistic model of spread of infection on a homogeneous tree. The model is related to stochastic SIR model on graphs. We derive an integral equation for the distribution of the waiting time for an individual to get infected. In a special case the integral equation is equivalent to the Bernoulli differential equation and implies the classic SIR model under an appropriate scaling.

In This paper we study certain fractional forms of Abel’s equation: $y'=P(x)+Q_{1}(x)y+Q_{2}(x)y^{2}+Q_{3}(x)y^{3}$. We solve the fractional form of the equation for the cases: $Q_{2} = 0$ or ${Q}_{3}= 0$. Such cases reduce the equation to Bernoulli fractional differential equation.

In this article we consider the two-point boundary value problem { u ( 4 ) ⁢ ( t ) = p ⁢ ( t ) ⁢ u ⁢ ( t ) + h ⁢ ( t ) for ⁢ a ≤ t ≤ b , u ( i ) ( a ) = c 1 ⁢ i , u ( i ) ( b ) = c 2 ⁢ i ( i = 0 , 1 ) , \left\{\begin{aligned} &\displaystyle u^{(4)}(t)=p(t)u(t)+h(t)\quad\text{for }% a\leq t\leq b,\\ &\displaystyle u^{(i)}(a)=c_{1i},\quad u^{(i)}(b)=c_{2i}\quad(i=0,1),\end{% aligned}\right. where c 1 ⁢ i , c 2 ⁢ i ∈ R {c_{1i},c_{2i}\in R} , h , p ∈ L ⁢ ( [ a , b ] ; R ) {h,p\in L([a,b];R)} . Here we study the question of dimension of the space of nonzero solutions and oscillatory behaviors of nonzero solutions on the interval [ a , b ] {[a,b]} for the corresponding homogeneous problem, and establish efficient sufficient conditions of solvability for the nonhomogeneous problem.

In this article, a new method is introduced for finding the exact solution of the product form of parabolic equation with nonlocal boundary conditions. Approximation solution of the present problem is implemented by the Ritz–Galerkin method in Bernoulli polynomials basis. The properties of Bernoulli polynomials are first presented, then Ritz–Galerkin method in Bernoulli polynomials is used to reduce the given differential equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the techniques presented in this article for finding the exact and approximation solutions. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1143–1158, 2017

In this article, Bernoulli wavelet method has been developed to solve nonlinear fuzzy Hammerstein-Volterra integral equations with constant delay. This type of integral equation has a particular case the fuzzy variant of a mathematical model from epidemiology. Bernoulli wavelets have been generated by dilation and translation of Bernoulli polynomials. The properties of Bernoulli wavelets and Bernoulli polynomials are first presented. The present wavelet method reduces these integral equations to a system of nonlinear algebraic equations and again these algebraic systems have been solved numerically by Newton's method. Convergence analysis of the present method has been discussed in this article. Also the results obtained by present wavelet method have been compared with that of by B-spline wavelet method. Some illustrative examples have been demonstrated to show the applicability and accuracy of the present method.

In this paper, we study the Fourier series related to higher-order Bernoulli functions and give new identities for higher-order Bernoulli functions which are derived from the Fourier series of them.

An asymptotic theory developed for a second-order differential equation. We obtain the form of solutions for some class of the coefficients for large x.

In the current study, new functions called generalized fractional-order Bernoulli wavelet functions (GFBWFs) based on the Bernoulli wavelets are defined to obtain the numerical solution of fractional-order pantograph differential equations in a large interval. For the concept of fractional derivative we will use Caputo sense by using Riemann-Liouville fractional integral operator. First, the generalized fractional-order Bernoulli wavelets are constructed. Then, these functions and their properties are employed to derive the GFBWFs operational matrices of fractional integration and pantograph. The operational matrices of integral and pantograph are utilized to reduce the problem to a set of algebraic equations. Finally, some examples are included for demonstrating the validity and applicability of our method.

Solution methods to exact differential equations via integrating factors have a rich history dating back to Euler (1740) and the ideas enjoy applications to thermodynamics and electromagnetism. Recently, Azevedo and Valentino presented an analysis of the generalized Bernoulli equation, constructing a general solution by linearizing the problem through a substitution. The purpose of this note is to present an alternative approach using ‘exact methods’, illustrating that a substitution and linearization of the problem is unnecessary. The ideas may be seen as forming a complimentary and arguably simpler approach to Azevedo and Valentino that have the potential to be assimilated and adapted to pedagogical needs of those learning and teaching exact differential equations in schools, colleges, universities and polytechnics. We illustrate how to apply the ideas through an analysis of the Gompertz equation, which is of interest in biomathematical models of tumour growth.

A new method to solve stationary one-dimensional Schroedinger equation is investigated. Solutions are described by means of representation of circles with multiple winding number. The results are demonstrated using the well-known analytical solutions of the Schroedinger equation.

New exact solutions of the Fractional Riccati Differential equation y(α) = a ( x) y2 + b ( x ) y + c ( x ) are presented. Exact solutions are obtained using several methods, firstly by reducing it to second order linear ordinary differential equation, secondly by transforming it to the Bernoulli equation, finally the solution is obtained by assuming an integral condition on c (x) involves an arbitrary function. Using the conditions imposed on Riccati equation's coefficients we choose the form of the coefficients of the Riccati equation. For this case the general solution of the Riccati equation is also presented.

In this paper, we define a new fractional function based on the Bernoulli wavelet to obtain a solution for systems of fractional differential equations (FDEs). The fractional derivative in these problems is in the Caputo sense. The method consists of expanding the required approximate solution as the elements of the fractional-order Bernoulli wavelets (FBWs). The fractional-order Bernoulli wavelets are constructed. The operational matrices of fractional order integration and derivative for FBWs are derived. The operational matrix of fractional integration and collocation method are utilized to reduce the initial value problems to system of algebraic equations. The error analysis shows that the method is convergent. Numerical examples are examined to highlight the significant features of the new technique.

In this note, we propose a generalization of the famous Bernoulli differential equation by introducing a class of nonlinear first-order ordinary differential equations (ODEs). We provide a family of solutions for this introduced class of ODEs and also we present some examples in order to illustrate the applications of our result.

Necessary and sufficient conditions which allow a second-order stochastic ordinary differential equation to be transformed to linear form are presented. The transformation can be chosen in a way so that all but one of the coefficients in the stochastic integral part vanish. The linearization criteria thus obtained are used to determine the general form of a linearizable Langevin equation.

In the present paper we propose a new proof of the Grosset–Veselov formula connecting one-soliton solution of the Korteweg–de Vries equation to the Bernoulli numbers. The approach involves Eulerian numbers and Riccati's differential equation.

In this paper, an analytical method based on the Bernoulli differential equation for extracting new complex soliton solutions to the Gilson&ndash, Pickering model is applied. A set of new complex soliton solutions to the Gilson&ndash, Pickering model are successfully constructed. In addition, 2D and 3D graphs and contour simulations to the complex soliton solutions are plotted with the help of computational programs. Finally, at the end of the manuscript a conclusion about new complex soliton solutions is given.

We investigate the realization of a Bernoulli-type first-order differential equation with a variable exponent. Using substitution methods, we show the existence of an implicit solution to the Bernoulli problem. Numerical simulations applied to several examples are also provided.

In this article, the generalized and improved (G′/G)-expansion method has been proposed for further extension to generate many new travelling wave solutions. In addition, nonlinear ordinary differential equation is implemented as auxiliary equation including many parameters instead of linear ordinary differential equation. Moreover, the presentation of the travelling wave solutions is quiet new. The effectiveness and reliability of the method are shown by its application to the Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZKBBM) equation. Some of our generated solutions turned into some known solutions, when parameters consider specific values and others are new.

We determine asymptotic forms of slowly decaying positive solutions of quasilinear ordinary differential equations related to so-called p-Laplace equation.

In this article, an implicit hybrid method of order six is developed for the direct solution of second order ordinary differential equations using collocation and interpolation approach.To derive this method, the approximate solution power series is interpolated at the first and off-step points and its second and third derivatives are collocated at all points in the given interval.Besides having good numerical method properties, the new developed method is also superior to the existing methods in terms of accuracy when solving the same problems.

The present study considers the solutions of hyperbolic partial differential equations. For this, an approximate method based on Bernoulli polynomials is developed. This method transforms the equation into the matrix equation and the unknown of this equation is a Bernoulli coefficients matrix. To demostrate the validity and applicability of the method, an error analysis developed based on residual function. Also examples are presented to illustrate the accuracy of the method.

For an ordinary differential equation with constant coefficients, we find a general representation of solutions; by using it, we solve the Dirichlet and Neumann problems, obtain conditions for the unique solvability, and construct closed-form representations of solutions.

A recent research has found out the relationship between the Log-aesthetic Curve and Riccati differential equations. The obtained differential equations are Riccati, but also Bernoulli type. Based on Sato and Shimizu's derivation of the formula of ρ-shift generalized log-aesthetic curve (GLAC), we derive the formula of the κ-shift GLAC as a solution of a Bernoulli differential equation.

In this study, we obtain some new complex analytical solutions to the Kundu–Eckhaus equation which seems in the quantum field theory, weakly nonlinear dispersive water waves and nonlinear optics using improved Bernoulli sub-equation function method. After we have mentioned the general structure of improved Bernoulli sub-equation function method, we have successfully applied this method and then obtained some new complex hyperbolic and complex trigonometric function solutions. Two- and three-dimensional surfaces of analytical solutions have been plotted via wolfram Mathematica 9 version. At the end of this article, a conclusion has been submitted by mentioning important points founded in this study.