# Analytical Analysis of Fractional-Order Physical Models via a Caputo-Fabrizio Operator

Authors: Fatemah Mofarreh; A. M. Zidan; Muhammad Naeem; Rasool Shah; Roman Ullah; Kamsing Nonlaopon
Journal: Journal of Function Spaces
 (2021)

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## Abstract

This paper investigates a modified analytical method called the Adomian decomposition transform method for solving fractional-order heat equations with the help of the Caputo-Fabrizio operator. The Laplace transform and the Adomian decomposition method are implemented to obtain the result of the given models. The validity of the proposed method is verified by considering some numerical problems. The solution achieved has shown that the better accuracy of the suggested method. Furthermore, due to the straightforward implementation, the proposed method can solve other nonlinear fractional-order problems.


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## Body

## 1. Introduction

The fractional generalization of differential equations has proven to be an effective and precise tool for interpreting real-world problems. Several methods have been proposed in this direction, including Ross and Miller, Podlubny, Caputo, and Kilbas et al. Many scholars, including Das and Gupta, Srivastava et al., and Singh et al., investigated the vibration equation’s fractional extension using the Caputo fractional derivative. Due to the Caputo and other fractional derivatives’ singular kernel, mathematical models involving these fractional derivatives do not consider the entire memory. For describing nonlocal behavioral models, fractional calculus (FC) has become a common mathematical approach. Fractional derivatives have mathematically interpreted many physical problems in recent decades; these representations have produced excellent solutions in real-world modeling issues. Riesz, Coimbra, Hadamard, Riemann-Liouville, Grunwald-Letnikov, Weyl, Caputo, Caputo-Fabrizio, Atangana-Baleanu, among others, gave many basic definitions of fractional operators [1, 2]. A wide variety of nonlinear equations have been developed and commonly applied in numerous nonlinear physical sciences such as chemistry, biology, mathematics, and various branches of physics such as plasma physics, condensed matter physics, fluid mechanics, field theory, and nonlinear optics over the past few years. The exact result of nonlinear equations plays a crucial role in deciding the characteristics and behavior of physical processes, but as dealing with linear equations, it is impossible to find their exact results. Many effective techniques have been used to solved nonlinear FPDEs, for example, the homotopy perturbation transform technique [3, 4], the homotopy analysis transform technique [5, 6], reduced differential transform technique [7, 8], the finite element technique [9], the finite difference technique [10], q-homotopy analysis transform method [11], and so on.The heat equation was created in 1822 by Joseph Fourier. The analysis of Brownian motion is related to the heat equation. Robert Brown treated this motion. The interaction of particles suspended in a liquid (fluid or gas) with the liquid’s fast-moving atoms causes the erratic movement of the particles. Heat refers to the complex energy of particles that are being exchanged. Faster moving atoms in the atmosphere clash with the scheme’s dividers as emotional vitality is transferred from an increased environment to a cooler system, exchanging some of this energy with the system’s atoms and enabling them to travel faster [12–14]. The wave model is a critical partial differential equation for explaining waves in conventional nuclear physics, such as mechanical waves. The electromagnetic radiation equation is a significant partial differential equation that occurs in areas such as acoustics, electrostatics, and quantum mechanics. Sound, light, gravity, and even matter are examples of them (in the Klein Gordon equation, relativistic quantum mechanics). Alembert and Euler discovered the one-dimensional and three-dimensional wave equations. The results of heat equations have been used of many researcher of mathematics, such as the Adomian decomposition technique [15], the optimal homotopy asymptotic technique [16], the differential transform technique [17], the variational iteration technique [18], Bernstein polynomials with the operational matrix [19], the homotopy perturbation method (HPM) [20], apply the variational iteration technique and Yang-Laplace transformation of time-fractional heat equations [21], Elzaki transformation and the differential transformation technique for nonlinear wave equation [22], and the Aboodh decomposition method [23]. The Laplace Adomian decomposition method (LADM) is a simple and efficient technique for solving linear and nonlinear FPDEs. LADM combines the Laplace transform and Adomian decomposition technique. It is observed that the proposed form, unlike RK4, does not require a predefined declaration size. In comparison to other analytical techniques, this method needs less parameters, no discretization, and no linearization [24]. Modified decomposition method is also compared to ADM to analyze the FPDE result provided in [25]. The result of Kundu Eckhaus equations is introduced in [26], using ADTM. Multistep ADTM are applied to investigate fractional partial differential equations in [27]. Laplace Adomian decomposition method is also applied for the result of fractional-order Smoke model [28]. In this article, we implemented LADM for the investigation of fractional-order heat equations. The current method has a very straightforward and simple method. Absolute error is used to measure accuracy.


## 2. Definitions and Preliminary Concepts

Definition 1.
Ifυ∈H0,T, T>0, ρ∈0,1, then Caputo-Fabrizio of fractional derivative is expressed as
(1)0CFDIρυI=Mρ1−ρ∫0Iυ′ζKI,ζdζ.
Mρ is the normalization function with M0=M1=1. If υ∈H0,T,
(2)0CFDIρυI=Mρ1−ρ∫0IυI−υζζKI,ζdζ.Definition 2.
The fractional Caputo-Fabrizio integral due toρ∈0,1 is defined by
(3)0CFIIρυI=1−ρMρυI+ρMρ∫0Iυζdζ,I≥0.Definition 3.
The Laplace transformation of relation with Caputo-Fabrizio as(4)L0CFDIρ+MυI11−ρυm+ρILe−ρI/1−ρ=1s+ρ1−ρsm+1LυI+∑k=1msm−kgk0.
Ifm=0,1(5)L0CFDIρυI=sLυζs+ρ1−s,L0CFDIρ+1υI=s2Lυζ+sυ0−υ′0s+ρ1−s.


## 3. The General Methodology of the Present Method

In this section, the general result finds the FPDEs with the help of the Laplace Adomian decomposition method.(6)DCFIρυζ,I+Gυζ,I+Nυζ,I=qζ,I,ζ,I≥0,m−1<ρ<m,where DCFρ=∂ρ/∂tρ the Caputo operator ρ,m∈ℕ, where G is linear and N nonlinear term, and the source term is q.With initial condition(7)υζ,0=kζ.Using the Laplace transformation to Equation (6), we get
(8)LDCFρυζ,I+LGυζ,I+Nυζ,I=Lqζ,I,apply the Laplace differentiation transform property, we obtain as
(9)sLυζ,I−υζ,0s+ρ1−s=Lqζ,I−LGυζ,I+Nυζ,I,(10)Lυζ,I=kζs+s+ρ1−ssLqζ,I−s+ρ1−ssLGυζ,I+Nυζ,I.The LADM resultυζ,I shows the infinite-series:
(11)υζ,I=∑ȷ=0∞υȷζ,I,the nonlinear term can be solve by Adomian polynomials
(12)Nυζ,I=∑ȷ=0∞Aȷ,(13)Aȷ=1ȷ!dȷdλȷN∑ȷ=0∞λȷυȷλ=0ȷ=0,1,2⋯,putting Equation (11) and Equation (12) in Equation (10), we achieve as
(14)L∑ȷ=0∞υζ,I=kxs+s+ρ1−ssLqζ,I−s+ρ1−ssLG∑ȷ=0∞υȷζ,I+∑ȷ=0∞Aȷ.Implementing of the Laplace transformation,(15)Lυ0ζ,I=υx,0s+s+ρ1−ssLqζ,I,Lυ1ζ,I=−s+ρ1−ssLGυ0ζ,I+A0.Usually, we can written as(16)Lυȷ+1ζ,I=−s+ρ1−ssLGυȷζ,I+Aȷ,ȷ≥1.Applying inverse transform in Equation (16), we get
(17)υ0ζ,I=kζ,I,υȷ+1ζ,I=−L−1s+ρ1−ssLGυȷζ,I+Aȷ.


## 4. Numerical Implementation

### 4.1. Problem 1

Consider the one-dimensional fractional heat equation (26):
(18)∂ρυ∂Iρ−ζ22∂2υ∂ζ2=0,0<ρ≤1,I≥0,with initial condition
(19)υζ,0=ζ2.Using the Laplace Caputo-Fabrizio operator in Equation (18),
(20)sLυζ,I−υx,0s+ρ1−s=Lζ22∂2υ∂ζ2.Applying inverse Laplace transformation,(21)υζ,I=L−1υx,0s−s+ρ1−ssLζ22∂2υ∂ζ2.Implement the ADM processes, we achieve(22)υ0ζ,I=L−1υx,0s=L−1ζ2s,υ0ζ,I=ζ2,υȷ+1ζ,I=L−1s+ρ1−ssLζ22∂2υȷ∂ζ2,ȷ=0,1,2,⋯,forȷ=0(23)υ1ζ,I=L−1s+ρ1−ssLζ22∂2υ0∂ζ2,υ1ζ,I=L−1ζ2s+ρ1−ss2=ζ21−ρ+ρI.The subsequent terms are(24)υ2ζ,I=L−1s+ρ1−ssLζ22∂2υ1∂ζ2=ζ2L−1s+ρ1−sss+ρ1−ss2,υ2ζ,I=ζ21+ρ2−2ρ−2ρ2t+2ρt+I2ρ22,υ3ζ,I=L−1s+ρ1−ssLζ22∂2υ2∂ζ2=ζ2L−1s+ρ1−sss+ρ1−s2s3,υ3ζ,I=ζ23ρ−2ρ+1+ρ2I+ρ3I36−3ρ2ρ−1I22+3ρ2−3ρ+1−ρ3.⋮The LADM solution for Problem 1 is(25)υζ,I=υ0ζ,I+υ1ζ,I+υ2ζ,I+υ3ζ,I+υ4ζ,I⋯υζ,I=ζ21+1−ρ+ρI+1−ρ+ρI2−I2ρ22+IIρ−ρ2−ρ3−3ρ+ρ−121−ρ+16I3ρ3⋯,when ρ=1, then the LADM result is
(26)υζ,I=ζ21+I+I22+I36+I424⋯.The close form of the Problem 1 is(27)υζ,I=ζ2eI.Figure1 shows the exact and approximate solution graphs with the help of Caputo-Fabrizio operator of example 1. In Figure 2, the first 3D graph shows the different fractional-order of ρ, and the second 2D figure shows the different fractional-order of ρ. In Table 1, values of the error, however, are proved to be proportional to the divergence of exact solutions and corresponding to the uncertainty in the approximate solutions.Figure 1
The first graph shows the exact, and the second graph shows the solution of analytical method atρ=1 of Problem 1.Table 1
The solutions of example 1 table at different fractional-order ofρ.
IξAbsolute error (ρ=0.4)Absolute error (ρ=0.6)Absolute error (ρ=0.8)Absolute error (ρ=1)0.115.1298036000×10−038.5968820000×10−041.1526960000×10−051.4693000000×10−1021.0533254900×10−021.7251291000×10−032.3064850000×10−052.1224000000×10−0931.5936706200×10−022.5905700000×10−033.4602750000×10−052.7754000000×10−0942.1340157400×10−023.4560108000×10−034.6140640000×10−053.4285000000×10−0952.6743608600×10−024.3214516000×10−035.7678540000×10−054.0815000000×10−090.216.5526269000×10−031.2845443000×10−041.9524030000×10−058.9043500000×10−0821.3585147000×10−022.5823037000×10−033.9081030000×10−051.2243480000×10−0832.0617667100×10−033.8800631000×10−035.8638020000×10−051.5582620000×10−0842.7650187200×10−025.1778225000×10−037.8195020000×10−051.8921750000×10−0853.4682707200×10−026.4755819000×10−039.7752020000×10−052.2260880000×10−080.317.5239217000×10−031.6203247000×10−042.6503270000×10−059.9570730000×10−0821.5715901300×10−023.2621458000×10−035.3069340000×10−051.2801951000×10−0832.3907880800×10−024.9039669000×10−037.9635420000×10−051.5646829000×10−0843.2099860300×10−026.5457880000×10−031.0620149000×10−051.8491707000×10−0854.0291839800×10−028.1876092000×10−031.3276756000×10−052.1336585000×10−080.418.2762123000×10−031.9075950000×10−033.2874570000×10−055.7825882000×10−0921.7398405800×10−023.8455493000×10−036.5848290000×10−056.7463529000×10−0832.6520599300×10−025.7835036000×10−039.8822010000×10−057.7101176000×10−0843.5642792800×10−027.7214580000×10−031.3179574000×10−058.6738823000×10−0854.4764986200×10−029.6594122000×10−031.6476946000×10−059.6376470000×10−080.518.8947364000×10−032.1627817000×10−033.8817930000×10−042.3900000000×10−0821.8806520800×10−024.3652454000×10−037.7777130000×10−043.5300000000×10−0832.8718305200×10−026.5677091000×10−031.1673633000×10−054.6600000000×10−0843.8630089800×10−028.7701729000×10−031.5569554000×10−045.8000000000×10−0854.8541874400×10−021.0972636700×10−031.9465475000×10−046.9300000000×10−08Figure 2
The analytical solution figures of different fractional-order ofρ of Problem 1.


### 4.2. Problem 2

Consider the time-fractional two-dimensional heat equation (26):
(28)∂ρυ∂Iρ−y22∂2υ∂ζ2−ζ22∂2υ∂ψ2=0,0<ρ≤1,with the initial condition
(29)υζ,ψ,0=ψ2.Applying the Laplace transform of Equation (28), we get
(30)sLυζ,ψ,I−υζ,ψ,0s+ρ1−s=Lψ22∂2υ∂ζ2+ζ22∂2υ∂ψ2.Applying inverse Laplace transformation,(31)υζ,ψ,I=L−1υζ,ψ,0s+s+ρ1−ssLψ22∂2υ∂ζ2+ζ22∂2υ∂ψ2.Applying the ADM processes, we achieve as(32)υ0ζ,ψ,I=L−1υζ,ψ,0s=L−1ψ2s,υ0ζ,ψ,I=ψ2,∑ȷ=0∞υȷ+1ζ,ψ,I=L−1s+ρ1−ssLψ22∑ȷ=0∞υxxȷ+ζ22∑ȷ=0∞υyyȷ,forȷ=0(33)υ1ζ,ψ,I=L−1s+ρ1−ssLψ22∂2υ0∂ζ2+ζ22∂2υ0∂ψ2,υ1ζ,ψ,I=ζ21−ρ+ρI.The subsequent terms are(34)υ2ζ,ψ,I=L−1s+ρ1−ssLψ22∂2υ1∂ζ2+ζ22∂2υ1∂ψ2=ψ21+ρ2−2ρ−2ρ2I+2ρI+I2ρ22,υ3ζ,ψ,I=L−1s+ρ1−ssLψ22∂2υ2∂ζ2+ζ22∂2υ2∂ψ2,υ3ζ,ψ,I=ζ23ρ−2ρ+1+ρ2I+ρ3I36−3ρ2ρ−1I22+3ρ2−3ρ+1−ρ3,υ4ζ,ψ,I=L−1s+ρ1−ssLψ22∂2υ3∂ζ2+ζ22∂2υ3∂ψ2,υ4ζ,ψ,I=ψ2−4ρ−3ρ2+3ρ−1+ρ3I−2ρ3ρ−1I33+3I2ρ4−2ρ3+ρ2−4ρ3+1+ρ4−4ρ+6ρ2+ρ4I424.⋮The LADM solution for Problem 2 is(35)υζ,ψ,I=υ0ζ,ψ,I+υ1ζ,ψ,I+υ2ζ,ψ,I+υ3ζ,ψ,I+υ4ζ,ψ,I⋯υζ,ψ,I=ψ2+ζ21−ρ+ρI+ψ21+ρ2−2ρ−2ρ2I+2ρI+I2ρ22+ζ23ρ−2ρ+1+ρ2I+ρ3I36−3ρ2ρ−1I22+3ρ2−3ρ+1−ρ3+ψ2−4ρ−3ρ2+3ρ−1+ρ3I−2ρ3ρ−1I33+3I2ρ4−2ρ3+ρ2−4ρ3+1+ρ4−4ρ+6ρ2+ρ4I424+⋯The closed form of the above equation is(36)υζ,ψ,I=ζ2sinhI+ψ2coshI.Figure3 shows the exact and approximate solution graphs with the help of Caputo-Fabrizio operator of example 2. In Figure 4, the first 3D graph shows the different fractional-order ofρ, and the second 2D figure shows the different fractional-order ofρ.Figure 3
The first graph shows the exact, and the second graph shows the solution of analytical method atρ=1 of Problem 2.Figure 4
The analytical solution graph of different fractional-order ofρ of Problem 2.


### 4.3. Problem 3

Consider the time-fractional three-dimensional heat equation (26):
(37)∂ρυ∂Iρ−ζψz4−136ζ2∂2υ∂ζ2+ψ2∂2υ∂ψ2+z2∂2υ∂z2,0<ρ≤1,I≥0,the initial condition is
(38)υζ,ψ,z,0=0.Using the Laplace transformation of Equation (37),
(39)sLυζ,ψ,z,I−υζ,ψ,z,0s+ρ1−s=Lζψz4+L136ζ2∂2υ∂ζ2+ψ2∂2υ∂ψ2+z2∂2υ∂z2.Implementing inverse Laplace transformation(40)υζ,ψ,z,I=L−1υζ,ψ,z,0s+s+ρ1−ssLζψz4+L−1s+ρ1−ssL136ζ2∂2υ∂ζ2+ψ2∂2υ∂ψ2+z2∂2υ∂z2.Apply the ADM processes, we achieve(41)υ0ζ,ψ,z,I=L−1υζ,ψ,z,0s+s+ρ1−ssLζψz4,υ0ζ,ψ,z,I=L−1ζ4ψ4z4s+ρ1−ss2=ζ4ψ4z41−ρ+ρI,∑ȷ=0∞ζ,ψ,z,I=L−1s+ρ1−ssL136ζ2∑ȷ=0∞υxȷ+ψ2∑ȷ=0∞υyȷ+z2∑ȷ=0∞υzȷ,forȷ=0(42)υ1ζ,ψ,z,I=L−1s+ρ1−ssLζ2∂2υ0∂ζ2+ψ2∂2υ0∂ψ2+z2∂2υ0∂z2,υ1ζ,ψ,z,I=ζ4ψ4z41+ρ2−2ρ−2ρ2I+2ρI+I2ρ22.The subsequent terms are(43)υ2ζ,ψ,z,I=L−1s+ρ1−ssLζ2∂2υ1∂ζ2+ψ2∂2υ1∂ψ2+z2∂2υ1∂z2,υ2ζ,ψ,z,I=ζ4ψ4z43ρ−2ρ+1+ρ2I+ρ3I36−3ρ2ρ−1I22+3ρ2−3ρ+1−ρ3,υ3ζ,ψ,z,I=L−1s+ρ1−ssLζ2∂2υ1∂ζ2+ψ2∂2υ1∂ψ2+z2∂2υ1∂z2,υ3ζ,ψ,z,I=ζ4ψ4z4−4ρ−3ρ2+3ρ−1+ρ3I−2ρ3ρ−1I33+3I2ρ4−2ρ3+ρ2−4ρ3+1+ρ4−4ρ+6ρ2+ρ4I424.⋮The LADM solution for Problem 3 is(44)υζ,ψ,z,I=υ0ζ,ψ,z,I+υ1ζ,ψ,z,I+υ2ζ,ψ,z,I+υ3ζ,ψ,z,I+⋯υζ,ψ,z,I=ζ4ψ4z41−ρ+ρI+1+ρ2−2ρ−2ρ2I+2ρI+I2ρ22+3ρ−2ρ+1+ρ2I+ρ3I36−3ρ2ρ−1I22+3ρ2−3ρ+1−ρ3+−4ρ−3ρ2+3ρ−1+ρ3I−2ρ3ρ−1I33+3I2ρ4−2ρ3+ρ2−4ρ3+1+ρ4−4ρ+6ρ2+ρ4I424+⋯,when ρ=1, then the LADM solution is
(45)υζ,ψ,z,I=ζ4ψ4z4I+I22!+I33!+I44!⋯.This is the close form solution of the above Problem 3(46)υζ,ψ,z,I=eI−1ζ4ψ4z4.Figure5 shows the exact and approximate solution graphs with the help of Caputo-Fabrizio operator of example 3. In Figure 6, the 3D graph shows the different fractional-order of ρ. In Figure 7, 2D figures show the different fractional-order of ρ with respect to ζ and I.Figure 5
The analytical solution graph of different fractional-order ofρ of Problem 3.Figure 6
The first graph shows the exact, and the second graph shows the solution of analytical method atρ=1 of Problem 3.Figure 7
The analytical solution graph of different fractional-order ofρ of Problem 3.


## 4.1. Problem 1

Consider the one-dimensional fractional heat equation (26):
(18)∂ρυ∂Iρ−ζ22∂2υ∂ζ2=0,0<ρ≤1,I≥0,with initial condition
(19)υζ,0=ζ2.Using the Laplace Caputo-Fabrizio operator in Equation (18),
(20)sLυζ,I−υx,0s+ρ1−s=Lζ22∂2υ∂ζ2.Applying inverse Laplace transformation,(21)υζ,I=L−1υx,0s−s+ρ1−ssLζ22∂2υ∂ζ2.Implement the ADM processes, we achieve(22)υ0ζ,I=L−1υx,0s=L−1ζ2s,υ0ζ,I=ζ2,υȷ+1ζ,I=L−1s+ρ1−ssLζ22∂2υȷ∂ζ2,ȷ=0,1,2,⋯,forȷ=0(23)υ1ζ,I=L−1s+ρ1−ssLζ22∂2υ0∂ζ2,υ1ζ,I=L−1ζ2s+ρ1−ss2=ζ21−ρ+ρI.The subsequent terms are(24)υ2ζ,I=L−1s+ρ1−ssLζ22∂2υ1∂ζ2=ζ2L−1s+ρ1−sss+ρ1−ss2,υ2ζ,I=ζ21+ρ2−2ρ−2ρ2t+2ρt+I2ρ22,υ3ζ,I=L−1s+ρ1−ssLζ22∂2υ2∂ζ2=ζ2L−1s+ρ1−sss+ρ1−s2s3,υ3ζ,I=ζ23ρ−2ρ+1+ρ2I+ρ3I36−3ρ2ρ−1I22+3ρ2−3ρ+1−ρ3.⋮The LADM solution for Problem 1 is(25)υζ,I=υ0ζ,I+υ1ζ,I+υ2ζ,I+υ3ζ,I+υ4ζ,I⋯υζ,I=ζ21+1−ρ+ρI+1−ρ+ρI2−I2ρ22+IIρ−ρ2−ρ3−3ρ+ρ−121−ρ+16I3ρ3⋯,when ρ=1, then the LADM result is
(26)υζ,I=ζ21+I+I22+I36+I424⋯.The close form of the Problem 1 is(27)υζ,I=ζ2eI.Figure1 shows the exact and approximate solution graphs with the help of Caputo-Fabrizio operator of example 1. In Figure 2, the first 3D graph shows the different fractional-order of ρ, and the second 2D figure shows the different fractional-order of ρ. In Table 1, values of the error, however, are proved to be proportional to the divergence of exact solutions and corresponding to the uncertainty in the approximate solutions.Figure 1
The first graph shows the exact, and the second graph shows the solution of analytical method atρ=1 of Problem 1.Table 1
The solutions of example 1 table at different fractional-order ofρ.
IξAbsolute error (ρ=0.4)Absolute error (ρ=0.6)Absolute error (ρ=0.8)Absolute error (ρ=1)0.115.1298036000×10−038.5968820000×10−041.1526960000×10−051.4693000000×10−1021.0533254900×10−021.7251291000×10−032.3064850000×10−052.1224000000×10−0931.5936706200×10−022.5905700000×10−033.4602750000×10−052.7754000000×10−0942.1340157400×10−023.4560108000×10−034.6140640000×10−053.4285000000×10−0952.6743608600×10−024.3214516000×10−035.7678540000×10−054.0815000000×10−090.216.5526269000×10−031.2845443000×10−041.9524030000×10−058.9043500000×10−0821.3585147000×10−022.5823037000×10−033.9081030000×10−051.2243480000×10−0832.0617667100×10−033.8800631000×10−035.8638020000×10−051.5582620000×10−0842.7650187200×10−025.1778225000×10−037.8195020000×10−051.8921750000×10−0853.4682707200×10−026.4755819000×10−039.7752020000×10−052.2260880000×10−080.317.5239217000×10−031.6203247000×10−042.6503270000×10−059.9570730000×10−0821.5715901300×10−023.2621458000×10−035.3069340000×10−051.2801951000×10−0832.3907880800×10−024.9039669000×10−037.9635420000×10−051.5646829000×10−0843.2099860300×10−026.5457880000×10−031.0620149000×10−051.8491707000×10−0854.0291839800×10−028.1876092000×10−031.3276756000×10−052.1336585000×10−080.418.2762123000×10−031.9075950000×10−033.2874570000×10−055.7825882000×10−0921.7398405800×10−023.8455493000×10−036.5848290000×10−056.7463529000×10−0832.6520599300×10−025.7835036000×10−039.8822010000×10−057.7101176000×10−0843.5642792800×10−027.7214580000×10−031.3179574000×10−058.6738823000×10−0854.4764986200×10−029.6594122000×10−031.6476946000×10−059.6376470000×10−080.518.8947364000×10−032.1627817000×10−033.8817930000×10−042.3900000000×10−0821.8806520800×10−024.3652454000×10−037.7777130000×10−043.5300000000×10−0832.8718305200×10−026.5677091000×10−031.1673633000×10−054.6600000000×10−0843.8630089800×10−028.7701729000×10−031.5569554000×10−045.8000000000×10−0854.8541874400×10−021.0972636700×10−031.9465475000×10−046.9300000000×10−08Figure 2
The analytical solution figures of different fractional-order ofρ of Problem 1.


## 4.2. Problem 2

Consider the time-fractional two-dimensional heat equation (26):
(28)∂ρυ∂Iρ−y22∂2υ∂ζ2−ζ22∂2υ∂ψ2=0,0<ρ≤1,with the initial condition
(29)υζ,ψ,0=ψ2.Applying the Laplace transform of Equation (28), we get
(30)sLυζ,ψ,I−υζ,ψ,0s+ρ1−s=Lψ22∂2υ∂ζ2+ζ22∂2υ∂ψ2.Applying inverse Laplace transformation,(31)υζ,ψ,I=L−1υζ,ψ,0s+s+ρ1−ssLψ22∂2υ∂ζ2+ζ22∂2υ∂ψ2.Applying the ADM processes, we achieve as(32)υ0ζ,ψ,I=L−1υζ,ψ,0s=L−1ψ2s,υ0ζ,ψ,I=ψ2,∑ȷ=0∞υȷ+1ζ,ψ,I=L−1s+ρ1−ssLψ22∑ȷ=0∞υxxȷ+ζ22∑ȷ=0∞υyyȷ,forȷ=0(33)υ1ζ,ψ,I=L−1s+ρ1−ssLψ22∂2υ0∂ζ2+ζ22∂2υ0∂ψ2,υ1ζ,ψ,I=ζ21−ρ+ρI.The subsequent terms are(34)υ2ζ,ψ,I=L−1s+ρ1−ssLψ22∂2υ1∂ζ2+ζ22∂2υ1∂ψ2=ψ21+ρ2−2ρ−2ρ2I+2ρI+I2ρ22,υ3ζ,ψ,I=L−1s+ρ1−ssLψ22∂2υ2∂ζ2+ζ22∂2υ2∂ψ2,υ3ζ,ψ,I=ζ23ρ−2ρ+1+ρ2I+ρ3I36−3ρ2ρ−1I22+3ρ2−3ρ+1−ρ3,υ4ζ,ψ,I=L−1s+ρ1−ssLψ22∂2υ3∂ζ2+ζ22∂2υ3∂ψ2,υ4ζ,ψ,I=ψ2−4ρ−3ρ2+3ρ−1+ρ3I−2ρ3ρ−1I33+3I2ρ4−2ρ3+ρ2−4ρ3+1+ρ4−4ρ+6ρ2+ρ4I424.⋮The LADM solution for Problem 2 is(35)υζ,ψ,I=υ0ζ,ψ,I+υ1ζ,ψ,I+υ2ζ,ψ,I+υ3ζ,ψ,I+υ4ζ,ψ,I⋯υζ,ψ,I=ψ2+ζ21−ρ+ρI+ψ21+ρ2−2ρ−2ρ2I+2ρI+I2ρ22+ζ23ρ−2ρ+1+ρ2I+ρ3I36−3ρ2ρ−1I22+3ρ2−3ρ+1−ρ3+ψ2−4ρ−3ρ2+3ρ−1+ρ3I−2ρ3ρ−1I33+3I2ρ4−2ρ3+ρ2−4ρ3+1+ρ4−4ρ+6ρ2+ρ4I424+⋯The closed form of the above equation is(36)υζ,ψ,I=ζ2sinhI+ψ2coshI.Figure3 shows the exact and approximate solution graphs with the help of Caputo-Fabrizio operator of example 2. In Figure 4, the first 3D graph shows the different fractional-order ofρ, and the second 2D figure shows the different fractional-order ofρ.Figure 3
The first graph shows the exact, and the second graph shows the solution of analytical method atρ=1 of Problem 2.Figure 4
The analytical solution graph of different fractional-order ofρ of Problem 2.


## 4.3. Problem 3

Consider the time-fractional three-dimensional heat equation (26):
(37)∂ρυ∂Iρ−ζψz4−136ζ2∂2υ∂ζ2+ψ2∂2υ∂ψ2+z2∂2υ∂z2,0<ρ≤1,I≥0,the initial condition is
(38)υζ,ψ,z,0=0.Using the Laplace transformation of Equation (37),
(39)sLυζ,ψ,z,I−υζ,ψ,z,0s+ρ1−s=Lζψz4+L136ζ2∂2υ∂ζ2+ψ2∂2υ∂ψ2+z2∂2υ∂z2.Implementing inverse Laplace transformation(40)υζ,ψ,z,I=L−1υζ,ψ,z,0s+s+ρ1−ssLζψz4+L−1s+ρ1−ssL136ζ2∂2υ∂ζ2+ψ2∂2υ∂ψ2+z2∂2υ∂z2.Apply the ADM processes, we achieve(41)υ0ζ,ψ,z,I=L−1υζ,ψ,z,0s+s+ρ1−ssLζψz4,υ0ζ,ψ,z,I=L−1ζ4ψ4z4s+ρ1−ss2=ζ4ψ4z41−ρ+ρI,∑ȷ=0∞ζ,ψ,z,I=L−1s+ρ1−ssL136ζ2∑ȷ=0∞υxȷ+ψ2∑ȷ=0∞υyȷ+z2∑ȷ=0∞υzȷ,forȷ=0(42)υ1ζ,ψ,z,I=L−1s+ρ1−ssLζ2∂2υ0∂ζ2+ψ2∂2υ0∂ψ2+z2∂2υ0∂z2,υ1ζ,ψ,z,I=ζ4ψ4z41+ρ2−2ρ−2ρ2I+2ρI+I2ρ22.The subsequent terms are(43)υ2ζ,ψ,z,I=L−1s+ρ1−ssLζ2∂2υ1∂ζ2+ψ2∂2υ1∂ψ2+z2∂2υ1∂z2,υ2ζ,ψ,z,I=ζ4ψ4z43ρ−2ρ+1+ρ2I+ρ3I36−3ρ2ρ−1I22+3ρ2−3ρ+1−ρ3,υ3ζ,ψ,z,I=L−1s+ρ1−ssLζ2∂2υ1∂ζ2+ψ2∂2υ1∂ψ2+z2∂2υ1∂z2,υ3ζ,ψ,z,I=ζ4ψ4z4−4ρ−3ρ2+3ρ−1+ρ3I−2ρ3ρ−1I33+3I2ρ4−2ρ3+ρ2−4ρ3+1+ρ4−4ρ+6ρ2+ρ4I424.⋮The LADM solution for Problem 3 is(44)υζ,ψ,z,I=υ0ζ,ψ,z,I+υ1ζ,ψ,z,I+υ2ζ,ψ,z,I+υ3ζ,ψ,z,I+⋯υζ,ψ,z,I=ζ4ψ4z41−ρ+ρI+1+ρ2−2ρ−2ρ2I+2ρI+I2ρ22+3ρ−2ρ+1+ρ2I+ρ3I36−3ρ2ρ−1I22+3ρ2−3ρ+1−ρ3+−4ρ−3ρ2+3ρ−1+ρ3I−2ρ3ρ−1I33+3I2ρ4−2ρ3+ρ2−4ρ3+1+ρ4−4ρ+6ρ2+ρ4I424+⋯,when ρ=1, then the LADM solution is
(45)υζ,ψ,z,I=ζ4ψ4z4I+I22!+I33!+I44!⋯.This is the close form solution of the above Problem 3(46)υζ,ψ,z,I=eI−1ζ4ψ4z4.Figure5 shows the exact and approximate solution graphs with the help of Caputo-Fabrizio operator of example 3. In Figure 6, the 3D graph shows the different fractional-order of ρ. In Figure 7, 2D figures show the different fractional-order of ρ with respect to ζ and I.Figure 5
The analytical solution graph of different fractional-order ofρ of Problem 3.Figure 6
The first graph shows the exact, and the second graph shows the solution of analytical method atρ=1 of Problem 3.Figure 7
The analytical solution graph of different fractional-order ofρ of Problem 3.


## 5. Conclusions

In this paper, the fractional-order approximate results of heat equations are calculated via LADM with the Caputo-Fabrizio operator. The solutions have shown the highest agreement with the actual results for the examples. This method results for some problems have revealed the accuracy of the suggested technique. It is observed that the achieved analytical series solutions for the first few terms are very precise and converge very rapidly to the solutions of real physical problems. In the future, LADM can be applied to investigate the analytical result of other linear and nonlinear fractional partial differential equations, which are frequently implemented in engineering and science.



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