AN ANALYTICAL STUDY OF REACTION DIFFUSION, (3 + 1)-DIMENSIONAL DIFFUSION EQUATIONS USING CAPUTO FABRIZIO FRACTIONAL DIFFERENTIAL OPERATOR

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. Motivated by the memory features and property to portray substance heterogeneities and arrangement with various sizes of Caputo Fabrizio operator of fractional order to investigate hidden dynamics of several nonlinear differential systems. In the present work, we conduct analytical study and obtain numerical simulations of one, two and (3+1)-dimensional Caputo-Fabrizio reaction-diffusion equations. Hybrid Laplace transform-based iterative method is constructed to find approximate solutions of diffusion equations involving Caputo–Fabrizio derivative with the exponential kernel. We have also carried out comparative analysis between Caputo and Caputo Fabrizio fractional differential operator. Moreover, we have obtained absolute error between exact and approximate solutions. 2D and 3D plots are obtained to demonstrate the efficiency of method graphically.


INTRODUCTION
The idea of fractional calculus is appeared more than 324 years prior, yet as of late it has draw the consideration of numerous researchers working in several areas of applied sciences and mathematical modeling. From the most recent couple of decades, fractional calculus turns into the most incredible asset to depict the nonlinear phenomena more accurately of numerous complex mathematical models, because of its good properties, such as memory effect, nonlocality, hereditary. [1,2,3] Indeed, different models of mathematical physics involving classical derivative have been reached out to represent non-local effects. The purpose of formulating mathematical models using fractional differential equations is to improve and generalize several ordinary differential systems. Hence, demonstrating some real world phenomena using fractional derivative operator has fascinated the consideration of many researchers in the field of applied mathematics. [4,5,6,7].
Nevertheless, it is complicated to investigate exact analytical solutions for these models represented by nonlinear differential equations of fractional order. To beat such shortcoming, many researchers across the globe have explored broadly reliable and powerful computational techniques to approximate their solutions turns into an essential problem of examination. Hence, our goal is to derive computational technique associated with the particular problem under examination and which protects the significant characteristic of solution of intrest such as boundedness, positivity,symmetry, convexity, monotonicity, among other properties [8,9]. Some of the useful methods are Adomain decomposition method [10,11], finite difference method [12], iterative Laplace transform method [13,14], spectral collocation method [15], homotopy perturbation method [16] and many other [17,18,19,20] In this work, we consider a fractional linear and nonlinear differential equations with reaction.
The term reaction means a generalized type of the reaction laws associated with the Hodgkin-Huxley and Fisher's models [21,22], Normally, these equations emerge as description models of several evolution processes in diiferent fields of applied science [23,24]. Several authors obtained approximate solutions of fractional reaction-diffusion equations [25,26,27,28] Inspired by this literature, in the present work, we have obtained approximate solutions of one, two and (3 + 1)-dimensional nonlinear reaction-diffusion equations utilizing a novel and systematic procedure called as the iterative Laplace transform method (ILTM). The suggested method is a combination of iterative method (NIM) and the Laplace transform. Moreover, we have applied Caputo-Fabrizio operator commensurate with the exponential decay law. This derivative operator is reliable and more appropriate to obtain solutions of numerous mathematical models of physical systems. We have considered the generalized one, two and (3 + 1)-dimensional Caputo-Fabrizio reaction-diffusion equations as follows where source term is denoted by the function g and λ is the diffusion coefficient.
The remaining part of this paper is arranged as follows. In Sections 2, useful preliminary of fractional calculus is presented. In Section 3, general iterative Laplace transforms method and approximate solution pertaining to general fractional reaction diffusion equation is discussed.
In Sections 4, we center around the verification of existence and stability criteria by employing Picard successive approximation technique and fixed point theory of Banach. In Section 5, the efficiency of suggested technique is illustrated by applying it on some examples. Moreover, the numerical simulations are presented graphically and with the help of tables. Section 6 is conclusions.

PRELIMINARIES
In this section, we present some useful definitions and lemmas of fractional calculus.
Similar to Caputo derivative operator, the CF operator gives CF D κ t v(x,t) = 0, if v is a constant function.
The benefit of Caputo-Fabrizio operator is that there is no singularity for t = s in the new kernel as compared to Caputo operator In particular, we have

ITERATIVE LAPLACE TRANSFORM METHOD
In this section, a general nonhomogeneous Caputo-Fabrizio fractional differential equation is considered which is given as below where source term is denoted by g(x,t) and linear and non-linear operators are shown by R and N respectively.
Employing the Laplace transform (6) on both sides of (7) gives Next, we apply inverse laplace transform on (8) then we get where ω(x,t) is the term derived from source term.
Further, we use new iterative method to obtain infinite series solution. This method is introduced by Daftardar-Gejji and Jafari [19].
since R is linear, The decomposion of nonlinear operator N is given as In view of (10), (11) and (12), the equation (9) is equivalent to further, consider the recurrence relation as follows The approximate solution with p−term is given as The convergence condition of the above approximate solution is obtained in [31].
Rearranging, we obtain Next, applying inverse Laplace transform on equation (18) The obtained series solution is given by, The recursive formula by using initial conditions is obtained as below.
where the nonlinear term ηv shows exact recurring process. The fixed-point set on τ is denoted by F (τ). Moreover, τ has atleast one element such that ξ p converges to h ∈ F (τ). Let {ς p } ⊆ B and define z p = ς p+1 − g(τ, ς p ) . If lim p→∞ z p = 0 implies that lim p→∞ ς p = h, then the iteration method ξ p+1 = g(τ, ξ p ) is called as τ-stable. Comparably, we think about that, this sequence {ς p } has an upper bound.
This iteration is called as Picard's iteration and it is τ− stable, if all these criterias are fulfilled for ξ p+1 = τξ p .
Theorem 4.1. Consider a Banach space (B, · ) and define τ as self-map on B fulfilling where Theorem 4.2. Consider a self-map τ defined as Proof. Here, we will show that τ consists a fixed point. Hence, for all (p, q) ∈ N × N, we consider the following.
By applying norm on both sides of (23) and without loss of generality, we obtain Next, utilizing triangular inequality and simplifying further (24) we get, Each term of Equation (26) can be evaluated as follows Further, this follows The boundedness of v p (x,t) and v q (x,t) implies existence of two distinct positive constants, π, µ such that for all (x,t), Hence, applying triangular inequality on (28) and using above positive constants we get Simplifying (26), by using equations (27) and (29), we obtain where Hence, the self-mapping τ has a fixed point. This completes the proof.
Further, we prove that τ satisfies all the criterias in Theorem 3.1. Let (30) holds then using Thus, all the conditions in Theorem 3.2 are satisfied by τ. Therefore, τ is Picard τ− stable.

NUMERICAL SIMULATIONS AND DISCUSSION
In the section, we exhibit the applicability of iterative Laplace transform method using numerical and graphical simulations of Caputo Fabrizio fractional reaction-diffusion equations.

CONCLUSIONS
In this work, we have conducted analytical study of one, two and (3+1)-dimensional Caputo-Fabrizio reaction-diffusion equations using iterative Laplace transform method. Moreover, by utilizing Banach theorem, the existence and stability criteria for steady solutions have been demonstrated. The approximate series solutions obtained by this efficient approach shows a reasonable consent to control the significant effect of diffusion dynamics for the different time period. The adequacy of this procedure can be radically improved by lessening steps and computing more components. Also, Caputo-Fabrizio fractional operator and the methodology presented in this work shall be appropriate for modeling other real-world problems.