Solutions Of Differential Equations Research Articles

A new treatment of boundary conditions in PDE solution with Galerkin methods via Partial Integral Equation framework

We present a new mathematical framework for solution of Partial Differential Equations (PDEs), which is based on an exact transformation of the underlying PDE that removes the boundary constraints from the solution state and moves them into the dynamics of the equivalent transformed equation. The framework is based on a Partial Integral Equation (PIE) representation of a PDE or a system of PDEs, where Partial Integral Equation does not require boundary conditions on its solution state. The PDE-PIE framework allows for a development of a generalized and consistent treatment of boundary conditions in constructing spectrally convergent solution approximations to a broad class of linear PDEs with non-constant coefficients governed by non-periodic boundary conditions, including, e.g., Dirichlet, Neumann and Robin boundaries, among others. The significance of this result is that a solution to almost any linear PDE in a form of a function series approximation can now be systematically constructed, irrespective of the boundary conditions. Furthermore, in many cases, the resulting ODE system for the expansion coefficients can now be integrated analytically in time, which allows us to obtain solution approximations to a broad class of unsteady PDEs with unprecedented accuracy. We present several PDE solution examples in one spatial variable implemented with the developed PIE-Galerkin methodology using both analytical and numerical integration in time. We also present comparison of the PIE methods with some classical direct PDE solution methods, further demonstrating advantages and potential limitations of the PIE approach. The developed framework can be naturally extended to multiple spatial dimensions and, potentially, to nonlinear problems.

General closed‐form equations for amplitude of parallel coupled quadrature oscillator

SummaryIn this paper, for the first time, a new method with closed‐form analytical equations is presented to calculate the oscillation amplitude of fourth‐order oscillators, such as quadrature oscillators. This method is actually based on the general form of the differential equations describing the structure of the fourth‐order oscillators and finding a solution for the nonlinear differential equations governing this type of oscillator. The introduced method is a general method that is valid for all fourth‐order oscillators and is also independent of the oscillation frequency. Using the proposed method, complex and time‐consuming simulation tools will no longer be needed to calculate the oscillation amplitude in a steady state. Moreover, the presented closed‐form equations help the designers to understand the design compromises and design the oscillator for their specific and desired conditions. In addition, to evaluate the correctness of the presented equations, a comprehensive analysis of the oscillation amplitude of the quadrature oscillator in the steady state is performed, and a closed‐form equation is presented for the oscillation amplitude of the oscillator in the steady state, which is proposed for the first time in this paper. A comparison between the simulation results and theoretical computations confirms the validity of the proposed method.

The exact meromorphic solutions of some nonlinear differential equations

The exact meromorphic solutions of some nonlinear differential equations

Entire solutions of higher order differential equations with entire coefficients having the same order

Entire solutions of higher order differential equations with entire coefficients having the same order

On the Existence and Ulam Stability of BVP within Kernel Fractional Time

This manuscript, we establish novel findings regarding the existence of solutions for second-order fractional differential equations employing Ψ-Caputo fractional derivatives. The application of Banach’s fixed-point theorem (BFPT) ensures the uniqueness of the solutions, while Schauder’s fixed-point theorem (SFPT) is instrumental in determining the existence of these solutions. Furthermore, we assess the stability of the proposed equation using the Ulam–Hyers stability criterion. To illustrate our results, we provide a concrete example showcasing their practical implications.

The Peano–Sard theorem for Caputo fractional derivatives and applications

The classical Peano–Sard theorem is a very useful result in approximation theory, bounding the errors of approximations that are exact on sets of polynomials. A fractional version was developed by Diethelm for fractional derivatives of Riemann–Liouville type, which we here extend to fractional derivatives of Caputo type. We indicate some applications to quadrature and interpolation formulae. These results will be useful in the approximate solution of fractional differential equations involving Caputo-type operators, which are often said to be more natural for applications.

Exact Solutions of Nonlinear Partial Differential Equations Using the Extended Kudryashov Method and Some Properties

In this paper, we consider how to find new exact solutions for nonlinear partial differential equations using the extended Kudryashov method. This method mainly uses the Riccati equation and the Bernoulli equation where there are some underdetermined constant parameters. And we also use the concept of symmetry to study its reduction equation, Lie transformation group, self-adjointness, and conservation laws. This paper mainly studies the Boussinesq class and the shallow water wave equation in (1 + 1) dimensions and tries to find new exact solutions and symmetry properties of them.

Shifted Legendre method for solutions of Riccati type functional differential equations and application of RL circuit model

AbstractIn order to solve the Riccati type functional differential equations under mixed condition, this research suggests a collocation approach based on shifted Legendre polynomials. These equations are seen in physics and electronic sciences. By means of the evenly spaced collocation points, the shifted Legendre polynomials, their derivatives and matrix forms, the method is constructed. It reduces the problem into a system nonlinear algebraic equations. Error analysis is made. The method is applied to numerical examples and their results are compared with other methods from literature. And also, the method is applied to nonlinear RL electrical circuit model. The results show that the method is effective.

Boundedness Analysis of the Fractional Maximal Operator in Grand Herz Space on the Hyperplane

The primary purpose of this work was to prove the boundedness of the Fractional Maximal Operator in Grand Herz Spaces on the Hyperplane. Here, We defined Grand Herz Space in a continuous Case. For Simplicity, We divided our Problem into two theorems by taking two subsets of Hyperplane( ) as ( ) and its complement . We proved the boundedness of the Fractional Maximal Operator in Grand Herz Space on these two subsets of Hyperplane. We also defined the continuous Case of Grand Herz Space. We proved some results to use in our proof. We represented other terms this paper uses, i.e. the Hyperplane and Fractional Maximal operator. Our proof method relied on one of the corollaries we gave in this paper. We proved the condition to apply that corollary, and then by referring to this, we confirmed both of our theorems. This paper is helpful in Harmonic analysis and delivers ways to analyse the solutions of partial differential equations. The Problem of our discussion provides methods to study the properties of very complex functions obtained from different problems from Physics, Engineering and other branches of science. Solutions of nonlinear Partial Differential equations often resulted in such functions which required deep analysis. Our work helps check the boundedness of such types of functions.

A comparative study on the rheological properties of upper convected Maxwell fluid along a permeable stretched sheet

The objective of this paper is to examine the flow of a non-Newtonian Maxwell fluid induced by a permeable stretching sheet in motion within a porous medium. The research incorporates the Cattaneo-Christov heat flux model to study the heat transfer process. The utilization of the Cattaneo-Christov heat flux approach becomes relevant in scenarios involving materials with high thermal conductivity or during short time intervals. Consequently, the current investigation holds significant importance. It is assumed that the viscosity of the Maxwell fluid changes exponentially as the temperature changes. The modeling of the physical phenomena being investigated takes into account the effects of a magnetic field, thermal radiation, velocity, and thermal slip conditions. In this study, the viscous dissipation phenomenon is taken into account because it can have notable impacts on the temperature and viscosity of the fluid, and is known to play a crucial role in fluid flow phenomena. The equations developed to model fluid flow are transformed into nonlinear ordinary differential equations through the use of appropriate similarity transformations. The focus of the research revolves around investigating the numerical solution of ordinary differential equations accompanied by boundary conditions using the shooting technique. The findings are then showcased via tables and graphs and scrutinized in order to arrive at conclusions. Furthermore, the precision of the present findings was evaluated by contrasting the heat transfer rate with outcomes that were previously published. Based on the obtained outcomes, it can be concluded that both the Eckert number and thermal radiation have a comparable enhancing influence, whereas the thermal relaxation parameter and thermal slip parameter exhibit opposing effects.

Predicting Tipping Points in a Family of PWL Systems: Detecting Multistability via Linear Operators Properties

The study of dynamical systems is based on the solution of differential equations that may exhibit various behaviors, such as fixed points, limit cycles, periodic, quasi-periodic attractors, chaotic behavior, and coexistence of attractors, to name a few. In this paper, we present a simple and novel method for predicting the occurrence of tipping points in a family of Piece-Wise Linear systems (PWL) that exhibit a transition from monostability to multistability with the variation of a single parameter, without the need to compute time series, i.e., without solving the differential equations of the system. The linearized system of the model is analyzed, the stable and unstable manifolds are taken to be real vectors in space, and the changes suffered by these vectors as a result of the modification of the parameter are examined using such simple metrics as the magnitude of a vector or the angle between two vectors in space. The results obtained with the linear analysis of the system agree well with those obtained with the numerical resolution of the dynamical system itself. The work presented here is an extension of previous results on this topic and contributes to the understanding of the mechanisms by which a system changes its stability by fragmenting its basin of attraction. This, in turn, enriches the field by providing an alternative to numerical resolution to identify quantitative changes in the dynamics of complex systems without having to solve the differential equation system.

Analytical Solutions of Partial Differential Equations Modeling the Mechanical Behavior of Non-Prismatic Slender Continua

Non-prismatic slender continua are the prototypical models of many structural elements used in engineering applications, such as wind turbine blades and towers. Unfortunately, closed-form expressions for stresses and strains in such continua are much more difficult to find than in prismatic ones, e.g., the de Saint-Venant’s cylinder, for which some analytical solutions are known. Starting from a suitable mechanical model of a tapered slender continuum with one dimension much larger than the other tapered two, a variational principle is exploited to derive the field equations, i.e., the set of partial differential equations and boundary conditions that govern its state of stress and strain. The obtained equations can be solved in closed form only in a few cases. Paradigmatic examples in which analytical solutions are obtainable in terms of stresses, strains, or related mechanical quantities of interest in engineering applications are presented and discussed.

Positive periodic solutions of third-order neutral differential equations with delayed derivative terms

By applying fixed point index theory on cones, the existence results of positive 2π-periodic solutions are obtained for the third-order neutral differential equation with delayed derivative terms in nonlinearity(x(t)−cx(t−δ))‴+a(t)x(t)=f(t,x(t),x(t−τ0),x′(t−τ1),x″(t−τ2)),t∈R under the condition that the nonlinear term f satisfies superlinear or sublinear growth, where constants δ>0,|c|<1, a:R→(0,+∞) and f:R×[0,+∞)2×R2→[0,+∞) are continuous functions which are 2π periodic in t and constants τ0,τ1,τ2>0.

Solutions of fractional differential equations by using a blend of variational iteration method with Sumudu transform and application to price adjustment equations

The presence of delays in a mathematical model can improve its vitality and suitability in describing several phenomena. However, in the presence of delays, nonlinear fractional differential equations are more difficult to study. This paper presents the use of a blend of variational iteration method with Sumudu transform for solving delay differential equations with Caputo derivatives of fractional variable order. Moreover, the paper introduces delays into the price adjustment equations to propose new price adjustment models with more potential for vitality and suitability. The paper assigns suitable real values to the parameters for the graphical display and comparison of the obtained solutions. The paper presents the interactions that exist among the price, demand, supply and dependence of supply and demand on the price, which can be applied to estimate the equilibrium price.

Numerical solutions of an option pricing rainfall weather derivatives model

Weather derivatives are financial products used to cover non catastrophic weather events with a weather index as the underlying asset. In this work, we derive a rainfall weather derivative price model, based in the assumption that the rainfall dynamics follows a Ornstein-Uhlenbeck process. To calculate the price of the option we arrive at a two dimensional time dependent partial differential equation, where the spatial variables are the rainfall index and the total rainfall. Appropriate boundary conditions are suggested and they differ from the boundaries presented in literature in similar contexts. To compute the approximate solutions of the partial differential equation, we propose an explicit numerical method in order to deal efficiently with the different choices of the coefficients involved in the equation, that depend on the rainfall defice (or excess) and on the precipitation (amount of rain). Being an explicit numerical method, it will be conditionally stable and we discuss the stability region of the numerical method and its order of convergence. In the end we examine two test cases where the parameters of the model presented are estimated based on precipitation data.

Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2

Abstract Our purpose in this article is to describe the solutions of several product-type nonlinear partial differential equations (PDEs) ( a 1 u + b 1 u z 1 + c 1 u z 2 ) ( a 2 u + b 2 u z 1 + c 2 u z 2 ) = 1 , \left({a}_{1}u+{b}_{1}{u}_{{z}_{1}}+{c}_{1}{u}_{{z}_{2}})\left({a}_{2}u+{b}_{2}{u}_{{z}_{1}}+{c}_{2}{u}_{{z}_{2}})=1, and ( a 1 u + b 1 u z 1 + c 1 u z 2 ) ( a 2 u + b 2 u z 1 + c 2 u z 2 ) = e g , \left({a}_{1}u+{b}_{1}{u}_{{z}_{1}}+{c}_{1}{u}_{{z}_{2}})\left({a}_{2}u+{b}_{2}{u}_{{z}_{1}}+{c}_{2}{u}_{{z}_{2}})={e}^{g}, where g ( z ) g\left(z) is a nonconstant polynomial and a j , b j {a}_{j},{b}_{j} , and c j ( j = 1 , 2 ) {c}_{j}\left(j=1,2) are constants in C {\mathbb{C}} . The finite-order transcendental entire solution u u of the first equation is of the following forms: u ( z 1 , z 2 ) = ± 1 a 1 a 2 + η 0 e 1 D [ ( a 2 c 1 − a 1 c 2 ) z 1 + ( a 1 b 2 − a 2 b 1 ) z 2 ] , u\left({z}_{1},{z}_{2})=\pm \frac{1}{\sqrt{{a}_{1}{a}_{2}}}+{\eta }_{0}{e}^{\tfrac{1}{D}{[}\left({a}_{2}{c}_{1}-{a}_{1}{c}_{2}){z}_{1}+\left({a}_{1}{b}_{2}-{a}_{2}{b}_{1}){z}_{2}]}, or u ( z 1 , z 2 ) = 1 2 a 1 e Q ( z 1 , z 2 ) + 1 2 a 2 e − Q ( z 1 , z 2 ) + η 0 e 1 D [ ( a 2 c 1 − a 1 c 2 ) z 1 + ( a 1 b 2 − a 2 b 1 ) z 2 ] , u\left({z}_{1},{z}_{2})=\frac{1}{2{a}_{1}}{e}^{Q\left({z}_{1},{z}_{2})}+\frac{1}{2{a}_{2}}{e}^{-Q\left({z}_{1},{z}_{2})}+{\eta }_{0}{e}^{\tfrac{1}{D}{[}\left({a}_{2}{c}_{1}-{a}_{1}{c}_{2}){z}_{1}+\left({a}_{1}{b}_{2}-{a}_{2}{b}_{1}){z}_{2}]}, where D = b 1 c 2 − b 2 c 1 D={b}_{1}{c}_{2}-{b}_{2}{c}_{1} , η 0 ∈ C − { 0 } {\eta }_{0}\in {\mathbb{C}}-\left\{0\right\} , and Q ( z 1 , z 2 ) = − 1 D [ ( a 1 c 2 + a 2 c 1 ) z 1 − ( a 1 b 2 + a 2 b 1 ) z 2 ] + η 1 , η 1 ∈ C . Q\left({z}_{1},{z}_{2})=-\frac{1}{D}\left[\left({a}_{1}{c}_{2}+{a}_{2}{c}_{1}){z}_{1}-\left({a}_{1}{b}_{2}+{a}_{2}{b}_{1}){z}_{2}]+{\eta }_{1},\hspace{1em}{\eta }_{1}\in {\mathbb{C}}. The description of the forms of the solutions for these PDEs demonstrates that our results are some improvements of the previous results given by Liu, Cao, and Xu [L. Xu and T. B. Cao, Solutions of complex Fermat-type partial difference and differential-difference equations, Mediterr. J. Math. 15 (2018), 227], and [K. Liu and T. B. Cao, Entire solutions of Fermat type difference differential equations, Electron. J. Diff. Equ. 2013 (2013), No. 59, 1–10.]. Meantime, we list some examples to explain that the forms of solutions of our theorems are precise to some extent.

On approximation of solutions of stochastic delay differential equations via randomized Euler scheme

We investigate existence, uniqueness and approximation of solutions to stochastic delay differential equations (SDDEs) under Carathéodory-type drift coefficients. Moreover, we also assume that both drift f=f(t,x,z) and diffusion g=g(t,x,z) coefficient are Lipschitz continuous with respect to the space variable x, but only Hölder continuous with respect to the delay variable z. We provide a construction of randomized Euler scheme for approximation of solutions of Carathéodory SDDEs, and investigate its upper error bound. Finally, we report results of numerical experiments that confirm our theoretical findings.

Optimal performance of a tontine overlay subject to withdrawal constraints

AbstractWe consider the holder of an individual tontine retirement account, with maximum and minimum withdrawal amounts (per year) specified. The tontine account holder initiates the account at age 65 and earns mortality credits while alive, but forfeits all wealth in the account upon death. The holder wants to maximize total withdrawals and minimize expected shortfall at the end of the retirement horizon of 30 years (i.e., it is assumed that the holder survives to age 95). The holder controls the amount withdrawn each year and the fraction of the retirement portfolio invested in stocks and bonds. The optimal controls are determined based on a parametric model fitted to almost a century of market data. The optimal control algorithm is based on dynamic programming and the solution of a partial integro differential equation (PIDE) using Fourier methods. The optimal strategy (based on the parametric model) is tested out of sample using stationary block bootstrap resampling of the historical data. In terms of an expected total withdrawal, expected shortfall (EW-ES) efficient frontier, the tontine overlay dramatically outperforms an optimal strategy (without the tontine overlay), which in turn outperforms a constant weight strategy with withdrawals based on the ubiquitous four per cent rule.

On asymptotic behavior of solutions of stochastic differential equations in multidimensional space

Consider the multidimensional SDE dX(t) = a(X(t)) dt + b(X(t)) dW(t). We study the asymptotic behavior of its solution X(t) as t → ∞, namely, we study sufficient conditions of transience of its solution X(t), stabilization of its multidimensional angle X(t)/|X(t)|, and asymptotic equivalence of solutions of the given SDE and the following ODE without noise: dx(t) = a(x(t)) dt.

Numerical Treatment for the Solution of Stochastic Fractional Differential Equation Using Lerch Operational Matrix Method

Abstract The article aims to propose the Lerch operational matrix method to solve a stochastic fractional differential equation. In this approach, the Lerch polynomials have been used as a basis function. Then, the product operational matrix, integral operational matrix, stochastic operational matrix, and operational matrix of fractional integral based on the Lerch polynomials have been constructed. The main characteristic of this method is to reduce the stochastic fractional differential equation into a system of algebraic equations by using derived operational matrices and suitable collocation points. Moreover, the convergence and error analysis of the presented method is also discussed in detail. Additionally, the applicability of the proposed technique is also demonstrated by solving some examples. To confirm the accuracy and effectiveness of the suggested technique, a comparison between the results produced by the proposed method and those obtained by other methods has been provided.