The study of thermophysical properties in convective stream through porous permeable medium is highly commendable in numerous engineering problems. Porous permeable metallic and ceramic materials have a lot of industrials and bio medical situations and no doubt could become modern materials with their vast applications in fields. Several mathematical and investigational studies were carried out to deliver a profounder consideration of the transportation instrument for the temperature transferal confidential the porous permeable medium. Porous permeable media has widespread solicitations like catalytic bed reactors, improving dehydrating effectiveness, sieving, isolation, and dealing petroleum recovery. The thermionic conductive porous permeable substances are used to augment forced convective heat transfer problems in several technological problems like reactor designing, thermal components like exchangers, and parabolic shape solar plates heaters ^{1, 2, 3, 4, 5}. The rapid and fast expansion of electronic machinery has developed more noticeable modifications in several manufacturing, business, and ecological problems. This expansion did not happen without an accompanied progress in heat dissipation (cooling) approaches. Augmentation of natural convection has been and will continue to be fundamental in refining the demonstration of the heat dissipation apparatuses of integrated circuit based technology confined.
Fin or stretched surface is a nice operational component to augment heat transfer from the heated regions and surfaces. It has been extensively functional in atmospheric heat conditioning, automotive, heat transfer equipment, etc. The main objectives of these investigations are dropping the dimensional/geometrical perspectives and material price of finbased system ^{3, 4, 5, 6, 7, 8}. The enhancement of heat dissipation capabilities in extended surfaces or fins have been given preferences in research and development that has guided to utilization on the use of porous fins. The enhancement of heat dissipation capabilities in extended surfaces have been a theme of enthusiastic prominence that has directed to widespread investigation on the use of porous finned structures (PFS). Two factors that are most prominent in system are; one is permeability i.e. inter connectivity of pores and scale of pores in PFS and second is convective load. Ghasemi et al. ^{7, 8} had a preliminary investigation on heat transfer in fin. They studied heat transfer of fin as compared to the porous fin escalations heat transfer geometrical benefits for a definite area and shrinkages the material mass by eliminating the solid mass with holes and cavities. It is important to narrate that Kiwan and AlNimr ^{9} initially announced perception of employing PFS to augment heat dissipation. Also Kiwan ^{10} familiarized the Darcy approach to designate the solidgas/air interfaces in the PFS. Khaled ^{11} studied heat dissipation in the quadrangular permeable fin and determined that the PFS improved heat dissipation rate. Kiwan and Zeitoun ^{12} established that the cylindershaped permeable fin augmented thermionic transitions. Ghasemi et al. ^{13} associated heat transfer presentations of solid permeable fins with 60 thermal controlled heat transfers. Kiwan ^{14} assessed thermionic radiation possessions in heat dissipation in (PFS). E. Cuce and P. M. Cuce ^{15} efficaciously used HPM to evaluate fin efficacy and performance analysis of the quadrangular permeable fin. Numerical approach has been used by Ma et al. ^{16} to investigate thermal capacity of the convective radiative permeable fin. Moradi et al. ^{17} employed a HPM modelled convective capacity and radiation thermal dissipation in the moveable porouspermeable fin. Bhanja et al. ^{18} formulated and followed analytical model based approach to attain fin performance and maximum design geometrical values and constraints of a movable porousstructuredfin uncovered in convectiveradiative environment. Some other scientists like Das ^{19} approved the inverse results of convectionradiation base approach for cylindershaped porouspermeable fins. The factors of permeability along with porosity and internal heat evolved in the system are linked to many applications. The many physiothermal characteristics of porousstructures in metals and ceramics have been investigated by the features of the connected solid part and the fluids encompassed in the porous membranes and structures ^{20, 21, 22, 23, 24, 25, 26, 27, 28, 29}. Mohammad Hemmat Esfe et al.^{ }
In the fractional calculus, the concept of nonintegral order differentiation and integration has been used. Fractional calculus is the generalization of the classical calculus. From the last many years, many mathematicians, researchers and scientists have observed that the role of noninteger operators is very important in expressing the properties of physical phenomena. Many procedures and apparatus have been proficiently expounded by fractional differentiation and integrals. Additional relative study has been done between classical models and fractional models ^{20, 21, 22, 23, 24, 25}. OHAM technique is rapidly convergent as compared to other techniques. The reliability and effectiveness of OHAM has been shown in literature with several applications in different fields of sciences and engineering.
The paper is categorized in four sections. In Section 2; the governing equation based upon the heat transfer equation is formulated. In Section 3; Computational remarks for solution based on OHAM are given. Some numerical examples are considered in section 4. The findings are tabulated and graphical results are shown in this section.
A quadrangular fin contour has chosen and deliberated in this work, for investigating the behavior of convective effects, as shown in
Energy balance can be written as:
The system is translated in Darcy's porousstructure approach with total energy balance of the system is presented as
Whereas, the massflow rate of the convectivefluid streaming PFS is given as
And velocity as flowsteam is given as :
In this system, it is presumed that heat energy generated in the fin varies with temperature. Here the concept of energy balance is used for steady state condition,
Porous parameter,
In this section, we give some basic definitions of real valued functions.
Definition: A function of real value
Definition: RiemannLiouville sense integral operator of a function
Definition RiemannLiouville sense derivative operator of a function
Caputo sense derivative operator of a function
If
According to the OHAM algorithm ^{16, 17}. We shall extend this scheme for fractional heattransfer differential equation {FPDE} in the following steps.
Compose the fractional order heattransfer governing equation as
Where
Where
Make an optimal homotopy for time fractional order partial differential equation,
Where
Where
Extend
It has been cleared that the rate of convergence of the (13) depends upon auxiliary constants
If it is convergent at p=1, one has
Substituting eq. (14) in eq. (10) yields the following expression for residual:
(i) Construct the
R is a residual of the problem Heat Transfer Equation
(ii) Auxiliary constants
If
The use of auxiliary constants in equation (14), we can get the rapidly convergent approximate solutions.
In this section, consider the Fractional Order Heat Transfer Problem for {OHAM} algorithm. This represents the accuracy, validity and effectiveness of the extended OHAM algorithm.
Let us consider the following heat transfer fractional order model from equation (4)
We can make an optimal homotopy in above equation with OHAM algorithm as
In the following
In the following
Residual of equation 17, can be written as
For the solution of auxiliary constant
Provides the values of
And here after substitution in (27) and simplification, we get
The OHAM algorithm presented in sec3 for time fractional order heattransfer equation and explanation of the formulation in the examples of sec4, provides extremely valid results for the problems without any spatial discretization. While applying {OHAM}, there is no need to compute higher order solutions. Values of materials properties are taken from ^{33} as
In the following
In this case, fractional thermal analysis for variations in
For
After using the equation (28), we acquire the following_{ }
Selecting the real value of c_{1, i.e., $$ c_{1}=0.12403630242308192 $$; similarly other real values of c1 with different values of \alpha are presented in the Table 1.}


1.2 
0.30231843281528586 
1.4 
0.2653176835941392 
1.6 
0.20822199156438242 
1.8 
0.12403630242308192 
X 
α = 1.2 
α = 1.4 
α = 1.6 
α = 1.8 
0 
1 
1 
1 
1 
0.1 
0.9480000443327921 
0.9744596122996947 
0.9890112905216663 
0.9964779968864528 
0.2 
0.8805354729303191 
0.9325985127704808 
0.9666884619982394 
0.9877356728222831 
0.3 
0.8056661391001716 
0.8810957871735966 
0.9362704724185412 
0.9745546754809605 
0.4 
0.7255425885491176 
0.8221264088055824 
0.8990182999895816 
0.9572931322679625 
0.5 
0.6412705938564254 
0.756899701789916 
0.8556892451032061 
0.9361830941972867 
0.6 
0.5535380273282889 
0.6862099011026739 
0.8068081985378031 
0.9113942336267902 
0.7 
0.4628190271555963 
0.6106279788060684 
0.7527686413036655 
0.8830591604082674 
0.8 
0.3694624458992538 
0.5305887786027642 
0.6938807287596512 
0.8512860489570467 
0.9 
0.2737368905864366 
0.4464370708163681 
0.6303976646426208 
0.8161658241048135 
1 
0.1758562425476251 
0.35845446680108883 
0.5625315961546157 
0.7777766268225289 
In this case, fractional thermal analysis for variations in
For
_{1}, i.e.


0.2 
0.27621397703119466 
0.4 
0.2762325604405811 
0.6 
0.27625124401164086 
0.8 
0.2762700280567808 
X 




0 
1 
1 
1 
1 
0.1 
0.9643597756885842 
0.9642984524855956 
0.964237124943611 
0.9641757930335941 
0.2 
0.9059448841587329 
0.9057830512558182 
0.9056212069022244 
0.9054593510213241 
0.3 
0.8340756269469365 
0.8337901345235279 
0.8335046218997645 
0.8332190889404659 
0.4 
0.7517870611975742 
0.7513599816063354 
0.7509328717965738 
0.7505057315660677 
0.5 
0.6607667330642797 
0.6601830422797601 
0.659599310195507 
0.6590155365351437 
0.6 
0.5621229543328996 
0.5613695353105931 
0.5606160629792303 
0.5598625369820687 
0.7 
0.4566524855151215 
0.4557175921308103 
0.45478263259700935 
0.45384760647104794 
0.8 
0.3449621273880217 
0.34383505758968325 
0.34270790804418494 
0.34158067821786087 
0.9 
0.22753298821876722 
0.22620386775780854 
0.22487465325335587 
0.22354534407607263 
1 
0.10475804136196842 
0.10321767214673072 
0.10167719394084285 
0.10013660601494234 
In this case, fractional thermal analysis for variations in G is given.
For G=0.8
After using the equation (28) we acquire the following
Selecting the real value of
G 

0.4 
0.2643088748972952 
0.6 
0.2652334985094893 
0.8 
0.2661653446931716 
1 
0.26640762311402805 
X G= 0.4 G= 0.6 G= 0.8 G= 1 0 1 1 1 1 0.1 0.9577701097529843 0.9575628385183665 0.9574665950851987 0.9573709069861848 0.2 0.8885546514959832 0.8880076594284759 0.8877536714856259 0.8875011490783947 0.3 0.8033971951997115 0.8024322363555874 0.8019841715182473 0.8015386920594542 0.4 0.7058939620602154 0.7044504413398036 0.7037801631401407 0.7031137525121626 0.5 0.598044515498646 0.596071651987834 0.595155581034706 0.594244795901826 0.6 0.4811620875717604 0.47861554591497624 0.47743309572015746 0.47625746838022875 0.7 0.3561907550807518 0.353030833606362 0.35156356926648746 0.3501047711873655 0.8 0.2238495126647626 0.22004003864486654 0.2182711638711189 0.21651249568023845 0.9 0.08470842265398071 0.08021602132018724 0.07813003901622739 0.07605609303703231 1 −0.06076688315617629 −0.06597330117495948 −0.06839082691392284 −0.07079440328962172
In this case, fractional thermal analysis for variations in m is given.
For
After using the equation (28) we acquire the following C_{1 }values
Selecting the real value of
m 

0.1 
0.20822199156438242 
0.2 
0.20952219162255087 
0.3 
0.21166422542123078 
0.4 
0.21461118137176743 
X 
m = 0.1 
m = 0.2 
m = 0.3 
m = 0.4 
0 
1 
1 
1 
1 
0.1 
0.9884704856329178 
0.9887225230142503 
0.988902915086611 
0.9890112905216663 
0.2 
0.9650490481399878 
0.965813082608519 
0.9663599291136072 
0.9666884619982394 
0.3 
0.9331340495163218 
0.9345957519370526 
0.9356419440830593 
0.9362704724185412 
0.4 
0.8940485265009441 
0.8963646459036193 
0.8980223745178205 
0.8990182999895816 
0.5 
0.8485870497179298 
0.8518969656670243 
0.8542659896357703 
0.8556892451032061 
0.6 
0.79730034223285 
0.8017313953816699 
0.804902856916581 
0.8068081985378031 
0.7 
0.7406012500646381 
0.7462717561738366 
0.7503303380515083 
0.7527686413036655 
0.8 
0.6788151927425872 
0.6858363538398891 
0.6908616472865354 
0.6938807287596512 
0.9 
0.612207835322971 
0.6206850459473411 
0.6267524855639679 
0.6303976646426208 
1 
0.5410017657465018 
0.551035554622101 
0.558217092373878 
0.5625315961546157 
In this case, fractional thermal analysis for variations in
For
After using the equation (28) we acquire the following C_{1 }values
Selecting the real value of
S_{h} 

2 
0.17993772776777064 
3 
0.26809839269109276 
4 
0.27625124401164086 
5 
0.2643088748972952 
X 




0 
1 
1 
1 
1 
0.1 
0.9573709069861848 
0.9642984524855956 
0.9739444712282638 
0.9882793793142147 
0.2 
0.8875011490783947 
0.9057830512558182 
0.9312390473326467 
0.9690690965718759 
0.3 
0.8015386920594542 
0.8337901345235279 
0.8786975289986707 
0.94543421999567 
0.4 
0.7031137525121626 
0.7513599816063354 
0.8185387580062936 
0.9183728564784756 
0.5 
0.594244795901826 
0.6601830422797601 
0.7519964501417977 
0.8884399713365505 
0.6 
0.47625746838022875 
0.5613695353105931 
0.6798808598348929 
0.8560000432535864 
0.7 
0.3501047711873655 
0.4557175921308103 
0.6027744754631792 
0.8213150943665177 
0.8 
0.21651249568023845 
0.34383505758968325 
0.5211209113812598 
0.784584676779108 
0.9 
0.07605609303703231 
0.22620386775780854 
0.43527189181477277 
0.7459670074391239 
1 
−0.07079440328962172 
0.10321767214673072 
0.34551470812468443 
0.7055913193050551 
In the above,
The above fractional thermal porous fin model, is a nonlinear secondorder ODE. In this work, the model solution is presented numerically by using the OHAM. It is presented that OHAM method offer unpretentious, correct and fitting method for forecasting heat dissipation effects in PFS with the occurrence of a thermionic system. We have two dissimilar types of cases subject to the tip situation of the fin, here the investigation of heat transfer is limited to porous fin of finitelength with insulated tip.
Constant 
Description 
A 
section area of fin 
OHAM 
Optimal Homotopy Asymptotic Method 
Cp 
Specific heat 
Ci 
Constants 
Da 
Darcy number 
G 
generation number, dimensionless 
H 
convection coefficient 
K 
thermal conductivity 
Kr 
thermal conductivity ratio [keff/kf] 
K 
Permeability 
L 
length of fin 
LS 
Least Square Method 
M 
convective parameter 
N 
number of iteration 
NUM 
Numerical Method 
Q 
conducted heat 
q* 
internal heat generation 
R(x) 
residual function 
Ra 
Rayleigh number 
Sh 
porous parameter 
T 
Temperature 
Tb 
temperature at fin base 
T∞ 
sink temperature for convection 