By Abdon Atangana

Derivative with a brand new Parameter: concept, equipment and purposes discusses the 1st program of the neighborhood by-product that used to be performed via Newton for common physics, and later for different components of the sciences.

The booklet begins by way of giving a heritage of derivatives, from Newton to Caputo. It then is going directly to introduce the hot parameters for the neighborhood spinoff, together with its definition and houses. extra issues outline beta-Laplace transforms, beta-Sumudu transforms, and beta-Fourier transforms, together with their houses, after which move directly to describe the tactic for partial differential with the beta derivatives.

Subsequent sections supply examples on how neighborhood derivatives with a brand new parameter can be utilized to version diverse functions, equivalent to groundwater circulation and varied ailments. The ebook offers an advent to the newly-established neighborhood spinoff with new parameters, besides their critical transforms and purposes, additionally together with nice examples on the way it can be utilized in epidemiology and groundwater studies.

- Introduce the recent parameters for the neighborhood by-product, together with its definition and properties
- Provides examples on how neighborhood derivatives with a brand new parameter can be utilized to version diversified functions, equivalent to groundwater move and varied diseases
- Includes definitions of beta-Laplace transforms, beta-Sumudu transforms, and beta-Fourier transforms, their homes, and strategies for partial differential utilizing beta derivatives
- Explains how the recent parameter can be utilized in a number of equipment

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**Additional resources for Derivative with a new parameter : theory, methods and applications**

**Example text**

26) 10. Distance scaling: We shall present in detail the proof of the above properties. 52 Derivative with a New Parameter Proof. Linearity: By definition, we have ∞ L(af(x) + bg(x))(s) = 0 ∞ =a 1 (β) t+ 0 ∞ +b β−1 1 (β) t+ t+ 0 e−st (af(t) + bg(t)) dt β−1 1 (β) e−st f(t) dt β−1 e−st g(t) dt = aL(f) + bL(g). 27) This completed the proof. Proof. 28) but A β 0 Dt (f(t − a)δ(t − a)) f t − a + ε t−a+ = lim 1 (β) 1−β · δ t−a+ε t−a + 1 (β) 1−β − f(t − a)δ(t − a) . 29) However, using the product rule for the beta-derivative, we obtain the following: A β 0 Dt (f(t−a)δ(t−a)) β β = δ(t−a)A0 Dt (f(t−a))+f(t−a)A0 Dt (δ(t−a)).

1. Let a ∈ R and g be a function, such that, g: [a, ∞) → R. 1) where g is a function such that g: [0, ∞) → R and the gamma-function ∞ (ζ ) = tζ −1 e−1 dt. 0 If the above limit of exists then g is said to be β-differentiable. Note that β for β = 1, we have A0 Dt g(t) = d g(t). Moreover, unlike other fractional dt Local derivative with new parameter 27 derivatives, the β-derivative of a function can be locally defined at a certain point, the same way like first-order derivative. 1. 5. Proof. Let f and g be differential and beta-differentiable on [a, b], let μ and α be two real number, then A 0 0 Dt g(t) = g(t).

3. The Sumudu transform is an integral transform similar to the Laplace transform, introduced in the early 1990s by Watugala [70] to solve differential equations and control engineering problems. The Sumudu transform of a function f, defined for all real numbers t 0, is the function Fs (u) , defined by: ∞ S(f(t)) = Fs (u) = 0 t 1 exp − f(t) dt. 4. The Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is often used in number theory, mathematical statistics, and the theory of asymptotic expansions; it is closely related to the Laplace transform and the Fourier transform, and the theory of the gamma function and allied special functions.