On Another Type of Transform Called Rangaig Transform

Abstract

A new Integral Transform was introduced in this paper. Fundamental properties of this transform were derived and presented such as the convolution identity, and step Heaviside function. It is proven and tested to solve some basic linear-differential equations and had succesfully solved the Abel's Generalized equation and derived the Volterra Integral Equation of the second kind by means of Initial Value Problem. The Natural Logarithm (e.g log ln e xx  ) has been established and defined by means of modifying the Euler Definite Integral based on the Rangaig's fomulation. Hence, this transform may solve some different kind of integral and differential equations and it competes with other known transforms like Laplace, Sumudu and Elzaki Transform.

The homotopy analysis Rangaig transform method for nonlinear partial differential equations

“…Accordingly, it can be said that the algorithm chosen through this combination is powerful and effective in its use in solving nonlinear partial differential equations and ordinary differential equation. Based on the paper [27], the author demonstrated that the Rangaig transform has a clearer and deeper connection to the Laplace transform. However, there are cases that the Laplace transform cannot solve the differential equations with the variable coefficients as 1 t , but can be solved by applying the Rangaig transform (see the example 4.3 and the example 4.4 of [27]).…”

“…Based on the paper [27], the author demonstrated that the Rangaig transform has a clearer and deeper connection to the Laplace transform. However, there are cases that the Laplace transform cannot solve the differential equations with the variable coefficients as 1 t , but can be solved by applying the Rangaig transform (see the example 4.3 and the example 4.4 of [27]). Therefore, the Rangaig transform can be used as an effective tool in solving integro-differential equations and is also effective when combining it with other methods to solve the linear and nonlinear in integer order or fractional order with variable coefficients of the form t n where n negative.…”

“…In this section, we give some basic definitions and properties of the Rangaig transform which are used further in this paper. A new transform called the Rangaig transform defined for a function of exponential order, we consider functions in the set H [27]:…”

“…In this transform, the variable ω factorizes the variable t of the function h or in another sense, the function h(t) is mapped into the function T (ω) of ω-space. We will summarize some results of the Rangaig transform for some functions [27,28] in Table 1.…”

“…The aim of this present study is to integrate two powerful methods for solving nonlinear partial differential equations, namely the homotopy analysis method and the Rangaig transformation method [27,28], to develop a new method. The Rangaig transform homotopy analysis method (HARTM) is the name of the updated method.…”