Improved pedagogy for linear differential equations by reconsidering how we measure the size of solutions

Tisdell, Christopher C.

doi:10.1080/0020739x.2017.1298856

Cited by 5 publications

(2 citation statements)

References 6 publications

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“…In this section we establish sufficient conditions on the existence of a unique solution for the boundary value problem (1.1) using Banach's fixed point theorem. "The field of fixed point theory aims to establish conditions under which certain classes of problems will admit one, or more, fixed points [21,20]" [16, C16]. First let us recall the statement of this theorem.…”

Section: Application Of Banach's Theoremmentioning

confidence: 99%

Existence and uniqueness of solutions to discrete,third-order three-point boundary value problems

Almuthaybiri

Jonnalagadda

Tisdell

2021

Cubo

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The purpose of this article is to move towards a more complete understanding of the qualitative properties of solutions to discrete boundary value problems. In particular, we introduce and develop sufficient conditions under which the existence of a unique solution for a third-order difference equation subject to three-point boundary conditions is guaranteed. Our contributions are realized in the following ways. First, we construct the corresponding Green's function for the problem and formulate some new bounds on its summation. Second, we apply these properties to the boundary value problem by drawing on Banach's fixed point theorem in conjunction with interesting metrics and appropriate inequalities. We discuss several examples to illustrate the nature of our advancements. RESUMENEl propósito de este artículo es avanzar hacia un entendimiento más completo de las propiedades cualitativas de las soluciones a problemas discretos de valor en la frontera. En particular, introducimos y desarrollamos condiciones suficientes bajo las cuales se garantiza la existencia de una única solución para una ecuación en diferencias de tercer orden sujeta a condiciones de borde en tres puntos. Nuestras contribuciones son de dos tipos. En primer lugar, construimos las funciones de Green correspondientes para el problema y formulamos nuevas cotas para su suma. En segundo lugar, aplicamos estas propiedades al problema de valor en la frontera usando el teorema del punto fijo de Banach junto con métricas interesantes y desigualdades apropiadas. Discutimos varios ejemplos para ilustrar la naturaleza de nuestros avances.

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Section: Application Of Banach's Theoremmentioning

confidence: 99%

Existence and uniqueness of solutions to discrete,third-order three-point boundary value problems

Almuthaybiri

Jonnalagadda

Tisdell

2021

Cubo

Self Cite

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“…Under this metric, the manifestations of familiar curves such as conic sections are found to exibit characteristics that both complement and contrast with their Euclidean counterparts [12], [13], [14]. Beyond such purely geometric territory, Tisdell [15] has recently suggested pedagogical benefits to approaches that use the taxicab metric when establishing a priori bounds on potential solutions to differential equations.…”

Section: Introductionmentioning

confidence: 99%

The taxicab locus of Apollonius: promoting exploratory routes to the punchline using rich undergraduate tasks

Miles

2018

International Journal of Mathematical Education in Science and

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A key motivational tactic in undergraduate mathematics teaching is to launch topics with fundamental questions that originate from surprising or remarkable phenomena. Nonetheless, constructing a sequence of tasks that promotes students' own routes to resolving such questions is challenging. This note aims to address this challenge in two ways. First, to illustrate the motivational tactic, the taxicab manifestation of a locus attributed to Apollonius is introduced and a natural question arising from comparison with the analogous Euclidean locus is considered, namely, does the taxicab locus of Apollonius ever coincide with a taxicab circle? Second, a companion sequence of rich undergraduate tasks is elaborated using theoretical design principles, with the tasks culminating in this fundamental geometric question. This note therefore provides a design approach that can be replicated in undergraduate teaching contexts based around similarly motivating mathematical phenomena.

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Pedogogical perspectives on the method of Wilmer III and Costa for solving second-order differential equations

Tisdell

2018

International Journal of Mathematical Education in Science and

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Improved pedagogy for linear differential equations by reconsidering how we measure the size of solutions

Cited by 5 publications

References 6 publications

Existence and uniqueness of solutions to discrete,third-order three-point boundary value problems

Existence and uniqueness of solutions to discrete,third-order three-point boundary value problems

The taxicab locus of Apollonius: promoting exploratory routes to the punchline using rich undergraduate tasks

Pedogogical perspectives on the method of Wilmer III and Costa for solving second-order differential equations

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