Analytic Methods for Solving Higher Order Ordinary Differential Equations

Abstract

In this work, an analytic approach for solving higher order ordinary differential equations (ODEs) is developed. The techniques offer analytic flexibility in many research areas such as physics, engineering, and applied sciences and are effective for solving complex ODEs.

Error Analysis Using Three and Four Stage Eighth Order Embedded Runge-Kutta Method for Sixth Order Ordinary Differential Equation vvi(u)=f(u,v,v',v'',v''',viv)

“…the local truncation errors [24][25][26] of v p (u) where p is a derivative, i.e., 0 p v = … are calculated after substitution of the exact solution of (1) into (11)(12)(13)(14)(15)(16).…”

“…Further, using equations (17)(18)(19) in equations (11)(12)(13)(14)(15)(16), the increment functions , , ,…”

“…For instance, third-order ODEs are used in thin film flow problems [7][8][9][10], and fifth-order differential equations are used in fiber preservation transformations [11]. Beccar et al [12] and Abdulsalam et al [13] contributed towards analyzing various orders of ODEs. Malhotra et al [14] optimized the real-life problem, and Kaur et al [15] used an improvised concept for solving the differential equations.…”

Analysis of implicit system of fractional order via generalized boundary conditions

“…The said system of Riemann–Liouville $$$ \mathcal{FDE} $$$ ()–() is the generalization of second‐order ordinary differential equation, and according to our information, the proposed ordinary differential equation have several uses in different engineering fields and the area of applied sciences [32–36]. Additionally, for $$$ \alpha =\beta =-1 $$$, we can acquire boundary conditions in the form of anti‐periodic, which often occur in the models of different physical phenomena (e.g., in the partial, ordinary and $$$ \mathcal{DE} $$$ with impulses, anti‐periodic trigonometric polynomials in the investigation of the interpolation issues, anti‐periodic wavelets, and equations in the discrete form) [37–41].…”

Adaptive Polynomial Method for Solving Third-Order ODE With Application in Thin Film Flow

“…Several engineering, chemical, science, and biological applications can be modeled, analyzed and/or solved using ordinary differential equations (ODEs). Simple first-order ODEs or more complex higher-order ODEs can be used to solve several problems in robotics, nuclear magnetic response, fluid equations, analysis of electrical circuits, and other applications [1][2][3]. The ODEs can be found in different forms including partial and algebraic problems [4,5].…”