Cauchy-Euler Differential Equation Research Articles

Closed-Form Solutions for Cauchy-Euler Differential Equations through the New Iterative Method (NIM)

Closed-Form Solutions for Cauchy-Euler Differential Equations through the New Iterative Method (NIM)

Investigation of the Boundary Layers of the Singular Perturbation Problem Including the Cauchy-Euler Differential Equation

‎In this paper‎, ‎for a singular perturbation problem consist of the Cauchy-Euler equation with local and non-local boundary conditions‎. ‎We investigate the condition of the self-adjoint and the non-self-adjoint‎, ‎also look for the formation or non-formation of boundary layers for local boundary conditions using the Frequent uniform limit method‎. ‎Also‎, ‎for the state of non-local conditions‎, ‎we convert the non-local boundary conditions into local conditions by finding the fundamental solution and then obtaining the necessary conditions with the help of the 4-step method‎. ‎Finally‎, ‎we determine the formation or non-formation of a boundary layer for non-local conditions such as local conditions‎.

On the Coefficients of Quasi-Convex Functions

In this paper we introduce and investigate the class of T (λ, β, A, B) that we call the class of quasi-starlike and quasi-convex functions with respect to the values of the parameter λ. Also we define the class K (λ, β, µ, m, A, B) which is satisfy non-homogeneous Cauchy-Euler differential equation. We obtain coefficient bounds for the functions belonging the above classes.

SOLUSI DARI PERSAMAAN CAUCHY–EULER NONHOMOGEN KASUS LOGARITMIK

Ordinary differential equation is one form of differential equations that are often found in everyday life. One form of ordinary differential equations which has non–constant coefficients is the Cauchy–Euler differential equation. In the nonhomogeneous Cauchy–Euler differential equations, the undetermined coefficient and the parameter variation were the most method that often used to find the particular solution. This paper aimed to show a new solution that was shorter than the previous methods for nonhomogeneous Cauchy–Euler differential equations with the right side was a logarithmic form. The new solution had been proven to produce the same solution as the ordinary solution sought using the undetermined coefficient method.

Penyelesaian Numerik Masalah Syarat Batas Robin pada Persamaan Diferensial Cauchy-Euler

This research studied how the numerical solution of the Cauchy-Euler differential equation with Robin boundary conditions. There were several numerical methods that can be used to get the numerical solution of a boundary value problem, namely the finite-difference method, the shooting method, the collocation method, and others. In this study, the numerical solution of Robin's boundary condition problem was obtained by the center finite-difference and the shooting methods. From the two methods, the numerical error was compared to the exact solution. The simulation results shown that the shooting method produces a better numerical solution for approximating the completion of the Cauchy-Euler differential equation than the finite-difference method since it produced smaller numerical errors.

Coefficients estimates for certain classes of analytic functions of complex order

In this paper, we introduce and investigate the subclasses $$\mathcal {Q} _{g}^{\zeta }(m,\lambda ,\mu ;\gamma )$$ and $$\mathcal {H}_{g}^{\zeta }(m,\lambda ,\mu ,r;u)$$ of analytic functions of complex order defined by a new differential operator involving binomial series. Furthermore, We obtain coefficients bounds and coefficient estimates involving of the nonhomogeneous Cauchy–Euler differential equation of order r. Several corollaries of the main results are also considered.

A General Method for the Ulam Stability of Linear Differential Equations

This paper deals with the Ulam stability of linear differential equations by using the method of variation of parameters, which provides a unified method to study the Ulam stability problem of linear differential equations of n-th order with constant and nonconstant coefficients. As an application of the main results, we also obtain the Hyers–Ulam stability of the Cauchy–Euler differential equations of second order, third order and n-th order. Our results make up to some deficiencies in the relevant literature.

Computing the two first probability density functions of the random Cauchy-Euler differential equation: Study about regular-singular points

Abstract In this paper the randomized Cauchy-Euler differential equation is studied. With this aim, from a statistical point of view, both the first and second probability density functions of the solution stochastic process are computed. Then, the main statistical functions, namely, the mean, the variance and the covariance functions are determined as well. The study includes the computation of the first and second probability density functions of the regular-singular infinite point via an adequate mapping transforming the problem about the origin. The study is strongly based upon the Random Variable Transformation technique along with some results that have been recently published by some of authors to the random homogeneous linear second-order differential equation. Finally, an illustrative example is shown.

Exact quantization of Cauchy-Euler type forced parametric oscillator

Driven and damped parametric quantum oscillator is solved by Wei-Norman Lie algebraic approach, which gives the exact form of the evolution operator. This allows us to obtain explicitly the probability densities, time-evolution of initially Glauber coherent states, expectation values and uncertainty relations. Then, as an exactly solvable model, we introduce the driven Cauchy-Euler type quantum parametric oscillator, which appears as self-adjoint quantization of the classical Cauchy-Euler differential equation. We discuss some typical behavior of this oscillator under the influence of external terms and give a concrete example.

Solution to a Class of First-Order Fuzzy Cauchy-Euler Differential Equations

Solution to a Class of First-Order Fuzzy Cauchy-Euler Differential Equations

The scaled boundary finite element method for the analysis of 3D crack interaction

The scaled boundary finite element method (SBFEM) can be applied to solve linear elliptic boundary value problems when a so-called scaling center can be defined such that every point on the boundary is visible from this scaling center. From a more practical point of view, this means that in linear elasticity, a separation of variables ansatz can be used for the displacements in a scaled boundary coordinate system. This approach allows an analytical treatment of the problem in the scaling direction. Only the boundary needs to be discretized with finite elements. Employment of the separation of variables ansatz in the virtual work balance yields a Cauchy–Euler differential equation system of second order which can be transformed into an eigenvalue problem and solved by standard eigenvalue solvers for nonsymmetric matrices. A further obtained linear equation system serves for enforcing the boundary conditions. If the scaling center is located directly at a singular point, elliptic boundary value problems containing singularities can be solved with high accuracy and computational efficiency. The application of the SBFEM to the linear elasticity problem of two meeting inter-fiber cracks in a composite laminate exposed to a simple homogeneous temperature decrease reveals the presence of hypersingular stresses.

Neighborhood properties of certain classes of multivalently analytic functions associated with the convolution structure

In this paper, by making use of the familiar concept of neighborhoods of p-valently analytic functions, we prove coefficient bounds, distortion inequalities and associated inclusion relations for the ( n, δ)-neighborhoods of a family of p-valently analytic functions and their derivatives, which is defined by means of a certain general family of non-homogenous Cauchy–Euler differential equations.

On neighborhoods of two new subclasses of multivalent functions with negative coefficients

In this paper, we introduce two new subclasses of p-valent analytic functions defined by means of fractional derivative of order δ . We obtained the various results including coefficient bounds and distortion inequalities for these function classes. Furthermore, we determine some inclusion relations for the ( n , p , ε ) -neighborhoods of a family of p-valent analytic functions with negative coefficients which is defined by means of a certain nonhomogeneous Cauchy–Euler differential equation.

Neighborhoods of certain p-valent analytic functions with complex order

In this paper, the authors prove several coefficient bounds, distortion inequalities of the class S g n ( p , λ , b , β ) of p-valent analytic functions of complex order. By making use of the familiar concept of neighborhoods of p-valent analytic functions, they obtained several inclusion relations associated with the N p ( n , δ ) -neighborhood of this class and its certain derivatives, which is defined by means of a certain non-homogeneous Cauchy–Euler differential equations.

The generalization of the generalized Al-Oboudi differential operator

In this paper, firstly we define the generalization of the generalized Al-Oboudi differential operator. Then we also define new classes of analytic and p-valently starlike and convex functions with complex order by means of this new general differential operator. Our main purpose is to determine coefficient bounds for functions in certain subclasses of this classes, which are introduced here by means of a family of nonhomogeneous Cauchy–Euler differential equations. Relevant connections of some of the results obtained with those in earlier works are also provided.

Neighborhoods of a certain class of p-valent functions with negative coefficients defined by using a differential operator

In this present paper, by making use of the familiar concept of neighborhoods of p valent functions, the author prove coefficient bounds and distortion inequalities, and associated inclusion relations for the (n, δ) -neighborhoods of a class of p -valently analytic functions with negative coefficients, which is defined by means of a certain non-homogeneous Cauchy-Euler differential equation. Relevant connections of some of the results obtained in this paper with those in earlier works are also provided. Mathematics subject classification (2000): 30C45.

Fractional Calculus and -Valently Starlike Functions

In this investigation, the authors prove coefficient bounds, distortion inequalities for fractional calculus of a family of multivalent functions with negative coefficients, which is defined by means of a certain nonhomogenous Cauchy-Euler differential equation.

Neighborhoods of a certain class of analytic functions with negative coefficients

A certain subclass of analytic functions in the open unit disc with negative coefficients is introduced. The new class is defined by means of multiplier transformations. By making use of the familiar concept of neighborhoods of analytic function, the author proves coefficient inequalities, distortion theorems and associated inclusion relations for the $(n,\delta)$-neighborhoods of functions belonging to the new class, which satisfy a certain nonhomogeneous Cauchy-Euler differential equation.

Certain subclass of analytic functions with complex order

In this paper, a subclass T ( n , λ , α , b ; μ ) of analytic functions with complex order is introduced by means of a family of nonhomogenous Cauchy–Euler differential equations. The main object of the paper is to determine coefficient bounds for the class, and obtain the exact results. These results extend some known theorems.

Coefficient bounds for some families of starlike and convex functions of complex order

In the present work, the authors determine coefficient bounds for functions in certain subclasses of starlike and convex functions of complex order, which are introduced here by means of a family of nonhomogeneous Cauchy–Euler differential equations. Several corollaries and consequences of the main results are also considered.