An Attempt to Make Non-Elementary Functions That Are Giving Solutions to Some Second-Order Nonlinear Autonomous ODEs

Stensland, Magne

doi:10.4236/jamp.2022.101006

Cited by 3 publications

(2 citation statements)

References 10 publications

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“…During the last 30 years there has been a lot of progress in finding solutions to nonlinear ODEs and PDEs. The progress is mostly made by using different methods like the Prelle-Singer method [3], Abel's equations [4] [5], the new Jacobi elliptic functions [6] [7], and the old Jacobi elliptic functions [8] [9], a new method [10], revised methods [11], Jacobi elliptic function expansion method [12], the expo-elliptic functions [13] [14], two new groups of non-elementary functions [15].…”

Section: Introductionmentioning

confidence: 99%

Six New Sets of the Non-Elementary Jef-Family-Functions that Are Giving Solutions to Some Second-Order Nonlinear Autonomous ODEs

Stensland¹

2023

JAMP

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In this paper, we define some new sets of non-elementary functions in a group of solutions x(t) that are sine and cosine to the upper limit of integration in a non-elementary integral that can be arbitrary. We are using Abel's methods, described by Armitage and Eberlein. The key is to start with a non-elementary integral function, differentiating and inverting, and then define a set of three functions that belong together. Differentiating these functions twice gives second-order nonlinear ODEs that have the defined set of functions as solutions. We will study some of the second-order nonlinear ODEs, especially those that exhibit limit cycles. Using the methods described in this paper, it is possible to define many other sets of non-elementary functions that are giving solutions to some second-order nonlinear autonomous ODEs.

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Section: Introductionmentioning

confidence: 99%

Six New Sets of the Non-Elementary Jef-Family-Functions that Are Giving Solutions to Some Second-Order Nonlinear Autonomous ODEs

Stensland¹

2023

JAMP

Self Cite

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show abstract

“…During the last 30 years there have been done a lot of progress in finding solutions to nonlinear ODEs and PDEs. The progress is mostly made by using different methods like the Prelle-Singer method [4], Abel's equations [5] [6], the new Jacobi elliptic functions [7] [8], the old Jacobi elliptic functions [9] [10], a new method [11], revised methods [12], Jacobi elliptic function expansion method [13], expo-elliptic functions [14].…”

Section: Introductionmentioning

confidence: 99%

Four New Examples of the Non-Elementary Expo-Elliptic Functions That Are Giving Solutions to Some Second-Order Nonlinear Autonomous ODEs

Stensland¹

2022

JAMP

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In this paper, we define four new examples of the non-elementary expo-elliptic functions. This is an exponential function whose exponent is the product of a real number and the upper limit of integration in a non-elementary integral that can be arbitrary. We are using Abel's methods, described by Armitage and Eberlein. We will study some of the second-order nonlinear ODEs, especially those that exhibit limit cycles, and systems of nonlinear ODEs that these functions are giving solutions to.

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On Two New Groups of Non-Elementary Functions That Are Giving Solutions to Some Second-Order Nonlinear Autonomous ODEs

Stensland¹

2022

JAMP

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An Attempt to Make Non-Elementary Functions That Are Giving Solutions to Some Second-Order Nonlinear Autonomous ODEs

Cited by 3 publications

References 10 publications

Six New Sets of the Non-Elementary Jef-Family-Functions that Are Giving Solutions to Some Second-Order Nonlinear Autonomous ODEs

Six New Sets of the Non-Elementary Jef-Family-Functions that Are Giving Solutions to Some Second-Order Nonlinear Autonomous ODEs

Four New Examples of the Non-Elementary Expo-Elliptic Functions That Are Giving Solutions to Some Second-Order Nonlinear Autonomous ODEs

On Two New Groups of Non-Elementary Functions That Are Giving Solutions to Some Second-Order Nonlinear Autonomous ODEs

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