G-CALCULUS. Keywords: Geometric real numbers; geometric arithmetic; bigeometric derivative.

Transcription

1 TWMS J. App. Eng. Math. V.8, N.1, 2018, pp CALCULUS K. BORUAH 1, B. HAZARIKA 1, Abstract. Based on M. rossman in 13] and rossman an Katz 12], in this paper we prove geometric Rolle s theorem, Taylor s theorem, Mean value theorem. Also, we discuss about the properties and applications of bigeometric calculus. Keywords: eometric real numbers; geometric arithmetic; bigeometric derivative. AMS Subject Classification: 26A06, 11U10, 08A05, 46A Introduction Non-Newtonian calculus also called as multiplicative calculus, introduced by rossman and Katz 12]. The operations of multiplicative calculus are called as multiplicative derivative and multiplicative integral. We refer to rossman and Katz 12], Stanley 20], Campbell 10], rossman 13, 14], Jane rossman 15, 16] for different types of Non-Newtonian calculus and its applications. Bashirov et al. 3] gaven the complete mathematical description of multiplicative calculus. An etension of multiplicative calculus to functions of comple variables found in 1, 2, 21, 22, 23]. The generalized Runge-Kutta method with respect to non-newtonian calculus studied by Kadak and Özlük 17]. Çakmak and Başar 7] constructed the field C of -comple numbers. Çakmak and Başar 8], the line and double integrals in the sense of -calculus are given. Moreover, in the sense of -calculus, the fundamental theorems of calculus for line integrals and double integrals are stated with some applications. Çakmak and Başar 9], characterized matri transformations in sequence spaces based on multiplicative calculus. Riza and Aktöre 18] discussed Runge-Kutta method in term of geometric multiplicative calculus. Bigeometric-calculus is one of the family of non-newton calculus. It provides differentiation and integration tools based on multiplication instead of addition. enerally, in growth related problems, price elasticity, numerical approimations problems Bigeometriccalculus can be advocated instead of a traditional Newtonian one. We refer 4, 6] to know basics of α generator and geometric arithmetic (R(,,,,. Türkmen and Başar 22] defined the sets of geometric integers, geometric real numbers and geometric comple numbers Z(, R( and C(, respectively, as follows: Z( = {e : Z}, R( = {e : R} = R + \{0}, C( = {e z : z C} = C\{0}. 1 Department of Mathematics, Rajiv andhi University, Rono Hills, Doimukh , Arunachal Pradesh, India. khirodb10@gmail.com; ORCID: bh rgu@yahoo.co.in; ORCID: Manuscript received: August 21, 2016; accepted: December 25, TWMS Journal of Applied and Engineering Mathematics, Vol.8, No.1 c Işık University, Department of Mathematics, 2018; all rights reserved. 94

2 -CALCULUS - K. BORUAH, B. HAZARIKA 95 If we take etended real number line, then R( = 0, ]. Remark 1.1. (R(,, is a field with geometric zero 1 and geometric identity e. But (C(,, is not a field, however, geometric binary operation is not associative in C(. For, we take = e 1/4, y = e 4 and z = e (1+iπ/2 = ie. Then ( y z = e z = z = ie but (y z = e 4 = e. eometric positive real numbers and negative real numbers are defined respectively as R + ( = { R( : > 1} and R ( = { R( : < 1} Useful relations between geometric and ordinary arithmetic operations. For all, y R( y = y y = /y y = ln y = y ln y or y = 1 ln y, y 1 2 = = ln p = lnp 1 = e (ln = e 1 ln e = and 1 = e n = n 2 = e y = e y y = y, if > 1 = 1, if = 1 1, if 0 < < 1 y y y = y y y 0 1 ( y = y. 2. Main Results 2.1. eometric Real Number Line. Consecutive geometric integers are geometrically equidistant as e n+1 e n = e n+1 n = e. Since (R(,, is a complete field with geometric identity e and geometric zero 1, so, we can consider a new number line with respect to geometric arithmetic which will be called geometric real number line eometric Co-ordinate System. We consider two mutually perpendicular geometric real number lines which intersect each other at (1, 1 as shown in FIURE 1 to form geometric co-ordinate system eometric Factorial. In 4], we defined geometric factorial notation! as n! = e n e n 1 e n 2 e 2 e = e n! eometric Pythagorean Triplets. Three numbers, y, z R( are said to be formed a geometric Pythagorean triplet(pt if 2 = y 2 z 2. (1 Or, equivalently ln = y ln y.z ln z or (ln 2 = (ln y 2 + (ln z 2. Thus, if {, y, z} R( is a PT, then {ln, ln y, ln z} forms an ordinary Pythagorean triplet(opt. Conversely, if {a, b, c} is a positive OPT, then {e a, e b, e c } forms a PT.

3 96 TWMS J. APP. EN. MATH. V.8, N.1, 2018 Y e 4 e 3 e 2 e X' e-4 e -3 e -2 e -1 1 e e 2 e 3 e 4 e -1 X e -2 e -3 e -4 Y' Figure 1. eometric Co-ordinate System Definition 2.1 (eometric Right Triangle. In the geometric co-ordinate system, if geometric lengths of the three sides of a triangle represent a PT, then the triangle will be called geometric right triangle. It is to be noted that a PT does not form a triangle in ordinary sense. Definition 2.2. Area of geometric right triangle = ln ( base altitude eometric Trigonometric Ratios. Let θ be an acute angle of a geometric right triangle and length of the sides be h, p, b R( with usual meaning. Then we define sing θ = p h = p 1 ln h cscg θ = h p = h 1 ln p cosg θ = b h = b 1 ln h secg θ = h b = h 1 ln b tang θ = p b = p 1 ln b cotg θ = b p = b 1 ln p Figure 2. eometric Right Triangle 2.6. Relation between geometric and ordinary trigonometry. Since n unit length in ordinary coordinate system is equal to e n unit in geometric coordinate system. So properties of the geometric right triangle having sides h, p, b R( will be same to the ordinary right triangle having sides h = ln(h, p = ln(p and b = ln(b, respectively. An eample is given in FIURE 2. Here, area of the both the triangles = 6 square unit, A = 36.87, B = and C = 90. It can be proved that sing θ = e sin θ, cosg θ = e cos θ, tang θ = e tan θ and sing θ cosg θ = tang θ.

4 -CALCULUS - K. BORUAH, B. HAZARIKA eometric Trigonometric Identities. We can verify that sing A cscg A = e, cos A sec A = e, sing 2 A cosg 2 A = e tang 2 A e = secg 2 A tang A cotg A = e, cotg 2 A e = cscg 2 A. sing(a + B = sing A cosg B cosg A sing B. cosg(a + B = cosg A cosg B sing A sing B Limit. According to rossman and Katz 12], geometric limit of a positive valued function defined in a positive interval is same to the ordinary limit. Here, we define -limit of a function with the help of geometric arithmetic as follows: A function f, which is positive in a given positive interval, is said to tend to the limit l > 0 as tends to a R, if, corresponding to any arbitrarily chosen number ɛ > 1, however small(but greater than 1, there eists a positive number δ > 1, such that 1 < f( l < ɛ for all values of for which 1 < a < δ. We write lim f( = l or f( l. Here, a a < δ a δ < < aδ and f( l < ɛ l ɛ < f( < lɛ. A function f is said to tend to limit l as tends to a from the left, if for each ɛ > 1 (however small, there eists δ > 1 such that f( l < ɛ when a/δ < < a. In symbols lim f( = l or f(a 1 = l. a Similarly, a function f is said to tend to limit l as tends to a from the right, if for each ɛ > 1, there eists δ > 1 such that f( l < ɛ when a < < aδ. In symbols lim f( = l or f(a + 1 = l. a Continuity. A function f is said to be -continuous at = a if (i f(a i.e., the value of f( at = a, is a definite number, (ii the -limit of the function f( as a eists and is equal to f(a. Alternatively, a function f is said to be -continuous at = a, if for arbitrarily chosen ɛ > 1, however small, there eists a number δ > 1 such that f( f(a < ɛ for all values of for which, a < δ. f( It is seen that a function f is -continuous at = a if lim a f(a = Basic Properties of -Calculus Derivative and its Interpretation. In 5] we defined the -differentiation of f( as ] d 1 f d = f f( h f( f(h ln h ( = lim = lim for h R(. (2 h 1 h h 1 f( The -derivative of a positive valued function f at a point c belonging to a positive interval can be defined as ] 1 f( f( f ln( c (c = lim = lim. (3 c c c

5 98 TWMS J. APP. EN. MATH. V.8, N.1, 2018 Equation (3 is the bigeometric slope defined by rossman in 13]. Depending on rossman 13], rossman and Katz 14], different researchers have been developing the bigeometric calculus taking arithmetic increment to the independent variable. But we are trying to develop their work with the help of geometric increments. So, to remove the confusion among the readers, instead of the phase bigeometric calculus the term -calculus is used throughout the paper. From (3, it is clear that -derivative eists if both f( and takes same sign and at the same time and c takes same sign. +h is arithmetic change and h = h is geometric change to the independent variable. Now, as changes to h, value of the function changes from f( to f( h = f(h. eometric changes to and y are given by = h = h = h and y = f( h f( = f(h f(. In case of ordinary derivative y h gives the average additive change in f( per unit change in over the interval, + ] =, + h]. Here in -calculus, ] 1 y = ( y 1 ln( f(h ln h = f( = f(+h f( gives the average geometric change in f( per unit geometric change in over the interval, h]. Now taking the limit as (i.e. h tends to 1, we get d y d = y lim 1 = lim 1 ( y 1 ln( ] 1 f(h ln h = lim. h 1 f( It is to be noted that -derivative eists if f( 0 and f(, f(h are both positive or both negative. y = m c i.e. y = c. ln m represents a straight line with slope m in geometric co-ordinate system as well as in log-log paper. Then, y = m. i.e. -derivative is the slope of the geometric straight line. Note: We ll denote n th geometric derivative by f n]. We call that left hand -derivative and right hand -derivative eist at = c if eist, respectively. lim c ( f(c.h 1 ln( c and lim c+ ( 1 f(c.h ln( c Theorem 3.1. If a function f is -differentiable and is positive, then it is both - continuous and ordinary continuous. Proposition 3.1. A continuous function f is not necessarily -derivable. Proof. Let us consider the function, if > 1 f( = = 1, if = 1, if 0 < < 1. 1 Then, obviously it is continuous at = 1. But it is not -differentiable at = 1 as Lf (1 = e but Rf (1 = 1 e. Eample 3.1. If f( = n, then f ( = e n (n 1 and f (n = e n!.

6 -CALCULUS - K. BORUAH, B. HAZARIKA 99 Remark 3.1. f( = n is a polynomial of degree n in ordinary sense, but geometrically it is a polynomial of degree one as n = e n. So, its -derivative is constant Relation between -derivative and ordinary derivative. By definition, - derivative of a positive valued function f( is given by f( h f( f ( = lim h 1 h ] 1 f(h ln h = lim, which is in 1 indeterminate form. h 1 f( Using logarithm, to transform it to 0 0 indeterminate form and then applying L Hospital rule, we can make a relation between -derivative and ordinary derivative as follows: f ( = lim e ln h 1 f(h f( ] 1 ln h = lim e ln f(h ln f( ln h h 1 = e lim h 1 hf (h f(h derivatives of some standard functions. -derivative of a constant: If f( = c, then f ( = 1 -derivative of ordinary product of a constant and a function: d cf ( (cf( = e cf( = e f ( f( = d d d (f(. -derivative of ordinary product of two functions: d = e f ( f(. (4 d d (f(.g( = (f(. (g(. (5 d d d -derivative of quotient of two functions: ( d f( d (f( d = d g( d (g(. (6 d -derivative of trigonometric functions: d d (sin = e cot, d d (cos = e tan, d d (tan = e sec csc, d sec csc (cot = e d d tan (sec = e d d d (csc = e cot. Theorem 3.2. If f : (a, b : R( is -differentiable, then (i f is increasing, if f 1. (ii f is decreasing, if f 1. Proof. Let c be an interior point of the domain a, b] of a function f and f (c eists ] 1 ln(/c. Then for given and be positive, i.e. f (c > 1. Then, f (c is the limit of ɛ > 1, δ > 1 such that ] f 1 (c f( ln(/c < ɛ If ɛ > 1 is so chosen that ɛ < f (c, then (i f( > 1, i.e. f( > if ]c, cδ, f( < ɛ.f (c where ]c/δ, cδ. ] 1 f( ln(/c > f (c ɛ > 1. Then

7 100 TWMS J. APP. EN. MATH. V.8, N.1, 2018 (ii f( < 1, i.e. f( < if ]c/δ, c. Thus from (i and (ii f( is increasing at = c. Hence the function is increasing at = c if f (c > 1. Similarly, it can be proved that the function is decreasing at = c if f (c < 1. Theorem 3.3 (eometric Darbou s Theorem. If a function f is -derivable on a closed interval a, b] and f (a, f (b are of opposite signs (i.e. one is > 1, other is < 1 then there eists at least one point c between a and b such that f (c = 0. Proof. Let f (a < 1 and f (b > 1. Since, -derivative eists ordinary derivative eists, so, f (a and f (b eist. Now f (a < 1 f (a < 0 and f (b > 1 f (b > 0. From Newtonian calculus, there eists c a, b] s.t. f (c = 0. So f (c = e c f (c = 1. Theorem 3.4 (eometric Intermediate value theorem for derivatives. If a function f is -derivable on a closed interval a, b] and f (a f (b and k be a number lying between f (a and f (b, then at least one point c ]a, b such that f (c = k. Proof. Let g( = f(. Then g (a = f (a ln k k so f (a k and g (b = f (b k. Since f (a < k < f (b, and f (b k can not be greater than 1 at the same time. Therefore, if g (a > 1 then g (b < 1. Hence, g( satisfies the conditions of Darbou s theorem. Thus, there eists at least one point c ]a, b such that g (c = 1, i.e. f (c = k. Theorem 3.5 (eometric Rolle s Theorem. If a function f defined on a, b] is (i -continuous on a, b], (ii -derivable on ]a, b, (iii f(a = f(b, then there eists at least one number c between a and b such that f (c = 1. Proof. Since -continuous functions are ordinary continuous and f ( eists if f ( eists. So, f satisfies the conditions of ordinary Rulle s theorem. So, there eists c ]a, b such that f (c = 0. Hence f (c = e cf (c = 1. Theorem 3.6 (Lagrange s Mean Value Theorem. If a function f defined on a, b] is (i -continuous on a, b], (ii -derivable on ]a, b, then there eists at least one c ]a, b such that f (c = f(b f(a ] 1 ln( b a Proof. Let us define a function φ( = ln k.f( where the constant k is so determined that φ(a = φ(b. a ] ln k φ(a = φ(b a ln k.f(a = b ln k f(b.f(b = b f(a. Using natural logarithm to both sides we get ] 1 ] f(b ln( a 1 b k = f(b ln( a = b. f(a f(a

8 -CALCULUS - K. BORUAH, B. HAZARIKA 101 Now, φ(, the product of two -derivable and -continuous functions, is itself (i -continuous on a, b], (ii -derivable on ]a, b, and (iii φ(a = φ(b. Therefore by eometric Rolle s theorem c ]a, b such that φ (c = 1. But φ ( = d d (ln k. d d (f( = k.f (. 1 = φ (c = k.f (c f (c = 1 k = f(b f(a ] 1 ln( b a. Note: If we replace b by ah, where h > 1, then c ]a, b may be taken as a.h ln θ for 1 < θ < e. Thus ] 1 f(ah f (a.h ln θ ln( = ah a f(a ln h f(ah = f(a. f (a.h ] ln θ, where 1 < θ < e. Now we deduce geometric Taylor s epansion for f(ah with the help of eometric Rolle s Theorem. Firstly, we have to find -derivative of two important functions as follows. Lemma 3.1. If y = f n] ( ] ln n ( ah n! then y = f n+1] (] ln n ( ah n! ln (n 1 ( ah f n] (] (n 1! Proof. Taking logarithm to both sides of y and differentiating, we get ( y d y = d f n] ( f n]. lnn ( ah + ln f n] (. lnn 1 ( ah. ( n! (n 1! e y y y = = e f n] ( f n]. ln n ( ah ( n!.e ln f n] (. ln (n 1 ( ah (n 1!. 1 ] lnn ( ah f n+1] ] ln (n 1 ( ah n! (. f n] (n 1! ( = ah 2 ah f n+1] ( ] lnn ( ah n! f n] ( ] ln(n 1 ( ah (n 1! Lemma 3.2. If y = k lnp ( ah where k is a constant and p is a positive integer, then y = k p ln(p 1 ( ah. Proof. Taking logarithm on the both sides, and then differentiating we get the result. Theorem 3.7 (eometric Taylor s Theorem. A function f defined on a, ah] is such that (i the (n 1 th -derivative of f, i.e. f n 1] is -continuous on a, ah], and (ii the n th -derivative, f n] eists on a, ah] then there eists at least one number θ between 1 and e such that ] ln h ] ln 2 h ] ln 3 h f(ah = f(a. f 1] (a. f 2] 2! (a. f 3] 3! (a... ] ln n 1 h ] (1 ln θ (n p ln n h... f n 1] (n 1! (a. f n] (a.h ln θ (n 1!p. (7

9 102 TWMS J. APP. EN. MATH. V.8, N.1, 2018 Proof. Condition (i in the statement implies that f 1], f 2], f 3],..., f n 1] eists and are continuous on a, ah]. Let us consider the function ] ln( ah φ( = f(. f 1] ( ] ln2 ( ah. f 2] ] lnn 1 ( ah 2! (... f n 1] (n 1! (.A lnp ( ah (8 where A is a constant to be determined such that φ(ah = φ(a. But, putting = ah and = a in (8, respectively, we get Now φ(ah = f(ah, and ] ln h ] ln 2 h ] ln n 1 h φ(a = f(a. f 1] (a. f 2] 2! (a... f n 1] (n 1! (a.a lnp h. ] ln h ] ln 2 h ] ln n 1 h f(ah = f(a. f 1] (a. f 2] 2! (a... f n 1] (n 1! (a.a lnp h. (9 (i f, f 1], f 2], f 3],..., f n 1] all being continuous on a, ah], the function φ( is continuous on a, ah]; (ii the functions f, f 1], f 2], f 3],..., f n 1] and ln r ( ah for all r being derivable in ]a, ah, the function φ( is derivable in ]a, ah; (iii φ(ah = φ(a. Hence, φ( satisfies all the conditions of Rolle s Theorem and hence there eists one real number θ ]1, e such that φ (a.h ln θ = 1. Now, using Lemma 3.1 and Lemma 3.2 f φ ( = f 1] 2] ( ] ln( ah f 3] ( ] ln2 ( ah 2! f n] (. f 1]. ( f 2] ( ]... ( ] ln ln( ah (n 1 ( ah (n 1! f n 1] ( ] ln(n 2 ( ah (n 2!.A p ln(p 1 ( ah which gives A = ] (1 ln θ (n p ln (n p h f n] (a.h ln θ (n 1!p. Now substituting the value of A in (9, we get ] ln h ] ln n 1 h ] (1 ln θ (n p ln n h f(ah = f(a. f 1] (a... f n 1] (n 1! (a. f n] (a.h ln θ (n 1!p. ( eometric Taylor s Series. In (10, the term R n = f n] (a.h ln θ (n 1!p is called Taylor s remainder after n terms. Since, 0 < 1 ln θ < 1 as 1 < θ < e, so, (1 ln θ n p 0 as n. Therefore, if f possesses -derivative of every order in a, ah] then R n 1 as n. Then Taylor s epansion becomes ] ln h ] ln n h ] ln n h f(ah = f(a. f 1] (a... f n] n! (a... = Π n=0 f n] n! (a. (11 This epression can be written in terms of geometric operations as f(a h = f(a h f 1] (a h2 2! f 2] (a... = n=0 ] (1 ln θ (n p ln n h h n n! f n] (a, (12

10 -CALCULUS - K. BORUAH, B. HAZARIKA 103 where h n = h ln(n 1 h. The equivalent epressions (11 and (12 will be called respectively as Taylor s product and eometric Taylor s series. If a, ah] then it also satisfies the conditions in the interval a, ]. Then replacing ah by or h by /a in (11, we get another form of Taylor s product as follows: ] ln( f( = f(a. f 1] a (a ] ln2 (. f 2] a ] lnn ( 2! (a... f n] a ] lnn ( n! (a... = Π n=0 f n] a n! (a. (13 4. Some applications of -calculus 4.1. Epansion of some useful functions in Taylor s product. (i. With the help of geometric Taylor s series, we can epress different functions as a product. For eample e = e.e ln.e ln2 2!.e ln3 3!... = e 1+ln + ln2 2! + ln3 3! +... (ii. -calculus gives better graphical and numerical approimations of functions than ordinary calculus. For, let f( = sin(. In the FIURE 3, we have given a comparison of linear approimation and eponential approimation respectively at = π 6. By ordinary Taylor s series, first order linear approimation is given by L( = f( π 6 + ( π 6 f ( π 6 = sin(π 6 + ( π 6 cos(π 6 = ( π By geometric Taylor s series, first order eponential approimation is given by E( = f( π 6. f 1] ( π 6 ] ln ( π/6 = sin( π 6. e π 6 cot( π ] ln( 6 π 1 ] 6 = 2. e π ln( 2 6 π 3. 2 sin( L( E( Figure 3. Eponential Approimation From the FIURE 3, it is clear that geometric Taylor s series gives better approimated value of the function f( = sin( at = π 6 than Taylor s approimation given by Michael Coco in 11] with the help of multiplicative derivative. (iii. -derivative gives total growth of a growth function. For, let y = a.b, where a =initial amount> 0, b = growth(or decay factor, =time and y =total amount after time period. Then, d y = b, which is the total growth or total decay according to b > 1 d or 0 < b < 1 respectively.

11 104 TWMS J. APP. EN. MATH. V.8, N.1, 2018 (iv. It is easy to find ordinary derivative of complicated product or quotient functions with the help of -derivative. For let, f( = e 1/2 n sin. Then f ( = d (e d 1/2 d ( d n. d (sin = d Therefore ordinary derivative is given by f ( = f( ln (f ( e2/2 e n.e cot = e 2 = e 1/2 n+1 sin 2 n cot ( 2 2 n cot. (v. Price Elasticity: With the aid of -derivative, we can find price elasticity to predict the impact of price changes on unit sales and to guide the firms profit-maimizing pricing decisions. According to 19, page no. 83], the price elasticity of demand is the ratio of the percentage change in quantity and the percentage change in the goods price, all other factors held constant. If and y represents price and quantity respectively, then the price elasticity E p is given by E p = % change in y % change in = y/y y / = y If price change is very small to the initially considered price, then making 0, we get E p = (e y y = ln y y = ln(y 1]. Resiliency = e (elasticity = e Ep = y 1]. 5. Conclusion Bigeometric calculus is one of the most actively discussed Non-Newtonian Calculus having variety of applications. Some of such applications and advantages are discussed in our papers 4, 5]. Seeing its importance, here we discussed different properties of bigeometric calculus using geometric increments to the independent variable. We have formulated basic identities which can be epressed in terms of geometric arithmetic independently. 6. Acknowledgment It is pleasure to thank Prof. M. rossman and Prof. Jane rossman for their constructive suggestions and inspiring comments regarding the improvement of the -calculus. References 1] A.E. Bashirov, M. Rıza, On Comple multiplicative differentiation, TWMS J. App. Eng. Math. 1(1( ] A. E. Bashirov, E. Mısırlı, Y. Tandoǧdu, A. Özyapıcı, On modeling with multiplicative differential equations, Appl. Math. J. Chinese Univ. 26(4( ] A. E. Bashirov, E. M. Kurpınar, A. Özyapici, Multiplicative Calculus and its applications, J. Math. Anal. Appl. 337( ] Khirod Boruah and Bipan Hazarika, Application of eometric Calculus in Numerical Analysis and Difference Sequence Spaces, arxiv: v1, 31 May ] Khirod Boruah and Bipan Hazarika, Some basic properties of -Calculus and its applications in numerical analysis, arxiv: v1, 24 July ] A. F. Çakmak, F. Başar, On Classical sequence spaces and non-newtonian calculus, J. Inequal. Appl. 2012, Art. ID , 12pages.

12 -CALCULUS - K. BORUAH, B. HAZARIKA 105 7] A.F. C akmak and F. Başar, Certain spaces of functions over the field of non-newtonian comple numbers, Abstr. Appl. Anal. 2014(2014, Article ID , 12 pages. 8] A.F. C akmak and F. Başar, On line and double integrals in the non-newtonian sense, AIP Conference Proceedings, 1611( ] A.F. C akmak and F. Başar, Some sequence spaces and matri transformations in multiplicative sense, TWMS J. Pure Appl. Math. 6(1( ] Duff Campbell, Multiplicative Calculus and Student Projects, Department of Mathematical Sciences, United States Military Academy, West Point, NY,10996, USA. 11] Michael Coco, Multiplicative Calculus, Lynchburg College. 12] M. rossman, R. Katz, Non-Newtonian Calculus, Lee Press, Piegon Cove, Massachusetts, ] M. rossman, Bigeometric Calculus: A System with a scale-free Derivative, Archimedes Foundation, Massachusetts, ] M. rossman, An Introduction to non-newtonian calculus, Int. J. Math. Educ. Sci. Technol. 10(4( ] Jane rossman, M. rossman, R. Katz, The First Systems of Weighted Differential and Integral Calculus, University of Michigan, ] Jane rossman, Meta-Calculus: Differential and Integral, University of Michigan, ] U. Kadak and Muharrem Özlük, eneralized Runge-Kutta method with respect to non-newtonian calculus, Abst. Appl. Anal., 2015 (2015, Article ID , 10 pages. 18] M. Riza and H. Aktöre, The Runge-Kutta Method in eometric multiplicative Calculus. LMS J. Comput. Math. 18(1( ] W.F. Samuelson, S.. Mark, Managerial Economics, Seventh Edition, ] D. Stanley, A multiplicative calculus, Primus IX 4 ( ] S. Tekin, F. Başar, Certain Sequence spaces over the non-newtonian comple field, Abstr. Appl. Anal. 2013(2013. Article ID , 11 pages. 22] Cengiz Türkmen and F. Başar, Some Basic Results on the sets of Sequences with eometric Calculus, Commun. Fac. Fci. Univ. Ank. Series A1. 61(2( ] A. Uzer, Multiplicative type Comple Calculus as an alternative to the classical calculus, Comput. Math. Appl. 60( Khirod Boruah is working as Research Scholar under the supervision of Bipan Hazarika at Rajiv andhi University, Arunachal Pradesh, India and he is studying on Sequence Spaces over non-newtonian Calculus and Numerical Methods over Bigeometric Calculus. Bipan Hazarika is working as Associate Professor at Rajiv andhi University, Arunachal Pradesh, India. He is studying on Summability Theory, Sequence Spaces, Functional Analysis, Non-Newtonian Calculus and Fuzzy Analysis. He has published more than 90 research articles in different international reputed journals.