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Existence and uniqueness of solutions is proved by Picard iteration. This is of particular interest since the proof actually tells us how to construct a sequence of functions that converge to our solution. 🔗 1.6.4 Reading Questions 🔗 1. Explain Theorem 1.6.1 in your own words. 🔗 2.

(ii) Under what conditions can we be sure that there is a unique solution to (*)? Here are the answers. Theorem 1 (Existence). Suppose that F(x, y) is a.

In 2019, Bae et al. [8] studied a theorem of existence and uniqueness of the solution to stochastic differential equations. Kim [1,2] considered ...

existence of a solution to equation (7) and hence our differential equation (1). Uniqueness There are several ways to prove the uniqueness of the solution of the initial value problem (1). None of them are difficult. We work in the interval [0, β] defined above. Say U~(t) and ~V (t) are both solutions. Let W~ (t) := U~(t) − V~ (t).

Request PDF | Existence of harmonic solutions for some generalisation of the non-autonomous Liénard equations | We study the problem of existence of harmonic solutions for some generalisations of ...

Existence and uniqueness theorem is the tool which makes it possible for us to conclude that there exists only one solution to a first order differential equation which satisfies a …

In mathematics – specifically, in differential equations – the Picard–Lindelöf theorem, Picard's existence theorem, Cauchy–Lipschitz theorem, or existence and ... A simple proof of existence of the solution is obtained by successive ...

01.01.2016 · Existence of solutions to fractional IVPs on time scales. In this section we prove the existence of a solution to the fractional order initial value problem , defined on a time scale. For this, let T be a time scale and J = [t 0, t 0 + a] ⊂ T. Then the function y ∈ C (J, R) is a solution of problem , if t 0 T D t α y (t) = f (t, y) on J, t ...

09.01.2022 · The recent paper considers a hydrodynamic flow of compressible biaxial nematic liquid crystal in dimension one. For initial density without vacuum states, we obtain both existence and uniqueness of global classical solutions. While for initial density with possible vacuum states, both the existence and uniqueness of global strong solutions are given.

We can have no solution, infinitely many solutions, or exactly one solution. Once again we would like to determine when there is one and only one solution of an ...

It was Emile Picard (1856–1941) who developed the method of successive approximations to show the existence of solutions of ordinary ...

Global Existence of Solutions. We assume that each admissible set of initial data for Δ is consistent with a solution defined on all of V.74 Uniqueness of Solutions. If Φ and Φ ′ are solutions that agree in the initial data that they induce on an instant ∑ ⊂ V, then they agree at any point x ∈ V at which they are both defined.

existence of a solution to equation (7) and hence our differential equation (1). Uniqueness There are several ways to prove the uniqueness of the solution of the initial value problem (1). None of them are difficult. We work in the interval [0, β] defined above. Say U~(t) and ~V (t) are both solutions. Let W~ (t) := U~(t) − V~ (t).

Existence and uniqueness theorem is the tool which makes it possible for us to conclude that there exists only one solution to a first order differential ...

20.11.2015 · 0 <2. We discuss the existence, uniqueness and stability of the weak solution. We also prove accurate estimates on the gradient of the solution near the boundary. Consequently, we can prove that the solution belongs to W1;q 0 for 1 <q< 1+ + 1 which is optimal if + >1. 1. Introduction In this article we study the quasilinear elliptic problem u ...

ODE: Existence and Uniqueness of a Solution. The Fundamental Theorem of Calculus tells us how to solve the ordinary differential equa- tion (ODE).

Jan 09, 2022 · The recent paper considers a hydrodynamic flow of compressible biaxial nematic liquid crystal in dimension one. For initial density without vacuum states, we obtain both existence and uniqueness of global classical solutions. While for initial density with possible vacuum states, both the existence and uniqueness of global strong solutions are given.

Motivated by the above works, the aim of this paper is to establish some sufficient conditions for the existence of nontrivial solution for the fractional differential equations (FDE) as follows. {Dαu(t) = f(t, v(t), Dνv(t)), t ∈ (0, T) u(0) = 0, u(T) = au(ξ), (3) where 1 < α < 2; ν, a > 0, ξ ∈ (0, T); α − μ ≥ 1 and Tα − 1 ...

(a) is an existence theorem. It guarantees that a solution exists on some open interval that contains x0, but provides no information on how to ...

Section 1.6 Existence and Uniqueness of Solutions. If \(x' = f(t, x)\) and \(x(t_0) = x_0\) is a linear differential equation, we have already shown that a solution exists and is unique. We will now take up the question of existence and uniqueness of solutions for all first-order differential equations. The existence and uniqueness of solutions will prove to be very important—even when we ...

15.01.2022 · About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy & Safety How YouTube works Test new features Press Copyright Contact us Creators ...

Existence and Uniqueness of Solution to ODEs: Lipschitz Continuity The study of existence and uniqueness of solution of ordinary differential equation (ODE) became important due to the lack of general formula for solving nonlinear ODEs. In this article, we shall discuss briefly about the existence and uniqueness of so-lution of a first order ...

Existence and Uniqueness of Solutions of Nonlinear Equations Although there are methods for solving some nonlinear equations, it’s impossible to find useful formulas for the solutions of most. Whether we’re looking for exact solutions or numerical approximations, it’s useful to know conditions that imply the existence and uniqueness of solutions of initial value problems for …