scholarly journals Newton's method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion

2017 ◽  
Vol 14 (1) ◽  
pp. 237-248
Author(s):  
Henryk Leszczyński ◽  
◽  
Monika Wrzosek
Keyword(s):  
Brownian Motion ◽  
Newton’S Method ◽  
Newton's Method ◽  
Wave Equations ◽  
One Dimensional ◽  
Stochastic Wave Equations

Newton’s method for nonlinear stochastic wave equations

2020 ◽  
Vol 32 (3) ◽  
pp. 595-605
Author(s):  
Henryk Leszczyński ◽  
Monika Wrzosek
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Wave Equations ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Order Convergence ◽  
Time Space ◽  
Stochastic Wave Equations ◽  
Second Order Convergence ◽  
Two Parameter

AbstractWe consider nonlinear stochastic wave equations driven by time-space white noise. The existence of solutions is proved by the method of successive approximations. Next we apply Newton’s method. The main result concerning its first-order convergence is based on Cairoli’s maximal inequalities for two-parameter martingales. Moreover, a second-order convergence in a probabilistic sense is demonstrated.

scholarly journals Converging Newton’s Method With An Inflection Point of A Function

2017 ◽  
Vol 13 (2) ◽  
pp. 73
Author(s):  
Ridwan Pandiya ◽  
Ismail Bin Mohd
Keyword(s):  
Iterative Method ◽  
Newton’S Method ◽  
Newton's Method ◽  
Inflection Point ◽  
Initial Point ◽  
One Dimensional ◽  
Root Finding ◽  
Inflection Points ◽  
Starting Point ◽  
The Relationship

For long periods of time, mathematics researchers struggled in obtaining the appropriate starting point when implementing root finding methods, and one of the most famous and applicable is Newton’s method. This iterative method produces sequence that converges to a desired solution with the assumption that the starting point is close enough to a solution. The word “close enough” indicates that we actually do not have any idea how close the initial point needed so that this point can bring into a convergent iteration. This paper comes to answer that question through analyzing the relationship between inflection points of one-dimensional non-linear function with the convergence of Newton’s method. Our purpose is to illustrate that the neighborhood of an inflection point of a function never fails to bring the Newton’s method convergent to a desired solution

scholarly journals Identification of the Point Sources in Some Stochastic Wave Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Yaozhong Hu ◽  
Guanglin Rang
Keyword(s):  
Point Source ◽  
Existence And Uniqueness ◽  
Wave Equations ◽  
Point Sources ◽  
Uniqueness Problem ◽  
One Dimensional ◽  
Stochastic Wave Equations ◽  
Dimensional Wave

We introduce and study a type of (one-dimensional) wave equations with noisy point sources. We first study the existence and uniqueness problem of the equations. Then, we assume that the locations of point sources are unknown but we can observe the solution at some other location continuously in time. We propose an estimator to identify the point source locations and prove the convergence of our estimator.

scholarly journals Converging Newton’s Method With An Inflection Point of A Function

2017 ◽  
Vol 13 (2) ◽  
pp. 73
Author(s):  
Ridwan Pandiya ◽  
Ismail Bin Mohd
Keyword(s):  
Iterative Method ◽  
Newton’S Method ◽  
Newton's Method ◽  
Inflection Point ◽  
Initial Point ◽  
One Dimensional ◽  
Root Finding ◽  
Inflection Points ◽  
Starting Point ◽  
The Relationship

For long periods of time, mathematics researchers struggled in obtaining the appropriate starting point when implementing root finding methods, and one of the most famous and applicable is Newton’s method. This iterative method produces sequence that converges to a desired solution with the assumption that the starting point is close enough to a solution. The word “close enough” indicates that we actually do not have any idea how close the initial point needed so that this point can bring into a convergent iteration. This paper comes to answer that question through analyzing the relationship between inflection points of one-dimensional non-linear function with the convergence of Newton’s method. Our purpose is to illustrate that the neighborhood of an inflection point of a function never fails to bring the Newton’s method convergent to a desired solution

Black Dots: Newton's Method and a Simple One-Dimensional Fractal

2001 ◽  
Vol 94 (9) ◽  
pp. 734-737
Author(s):  
Tony J. Fisher
Keyword(s):  
Differentiable Function ◽  
Newton’S Method ◽  
Newton's Method ◽  
Simple Complex ◽  
One Dimensional ◽  
Fractal Patterns ◽  
Fractal Images ◽  
Complex Valued

Students in a standard calculus course learn Newton's method for finding the root of a differentiable function. Although they may often see a diagram that visually demonstrates how this method works, it often soon becomes yet another algorithm to memorize or to program into a calculator. In addition, students are sometimes told that using Newton's method on simple complex-valued functions can lead to beautiful fractal patterns. However, the connection between the sequence of steps that they have learned and the corresponding fractal images is fuzzy at best. This article describes a calculator exercise that can help students develop a better visual and numeric feel for Newton's method and discover how Newton's method can lead to a simple, one-dimensional fractal.

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