Solution of Higher‐Order ODEs Using Backward Difference Method

Abstract

The current numerical technique for solving a system of higher-order ordinary differential equations (ODEs) is to reduce it to a system of first-order equations then solving it using first-order ODE methods. Here, we propose a method to solve higher-order ODEs directly. The formulae will be derived in terms of backward difference in a constant stepsize formulation. The method developed will be validated by solving some higher-order ODEs directly with constant stepsize. To simplify the evaluations of the integration coefficients, we find the relationship between various orders. The result presented confirmed our hypothesis.

The solution of Riccati type differential equation by means of variable order variable stepsize backward difference method

“…Problems 4-5 are Painleve's first and second transcendent in the form of Riccati differential equation with out exact solutions. Type of error used to estimate the solutions can be found in [14]. The notations below will indicate …”

Solving Directly Higher Order Ordinary Differential Equations by Using Variable Order Block Backward Differentiation Formulae

“…Real world problems from various applications in science and engineering can often be modeled into ordinary differential equations (ODEs). Some of the problems are modeled in the form of higher order ODEs [1] in such a way that ODE will describe the behavior of the problems. The main focus of this paper is on the linear third-order stiff initial value problems (IVPs).…”

“…The linear third order ODEs is categorized as higher order ODE. Define the third-order ODE with its initial conditions as y = f (x, y, y , y ), or rewrite it as y = αy + βy + γy equipped with initial conditions y(a) = y 0 , y (a) = y 0 , y (a) = y 0 (1) where x ∈ [a, z], a is the starting point and z is the end point.…”

A 4-Point Block Method for Solving Higher Order Ordinary Differential Equations Directly

“…Many natural processes or real-world problems can be translated into the language of mathematics [1][2][3][4]. The mathematical formulation of physical phenomena in science and engineering often leads to a differential equation, which can be categorized as an ordinary differential equation (ODE) and a partial differential equation (PDE).…”

“…Commonly, the formulation of real-world problems will take the form of a higher order differential equation associated with its initial or boundary conditions [4]. In the literature, a mathematical model in the form of a fifth-order differential equation, known as Korteweg-de Vries (KdV) equation, has been used to describe several wave phenomena depending on the values of its parameters [2,3,5,6].…”