Heat transfer in convection saturated porous medium have been initiated increasing interest in studying their significance in many applications in geophysical and ^{1}. The study of heat transfer in a free convective vertical flat plate immersed in a porous medium with an internal heat generation problem is solved by Cortell R with the aid of the Fourthorder RungeKutta algorithm ^{2}. This work is extended on the previous work of the authors Driss Achemlal et al. who have investigated the effect of heat source and thermal radiation flux around a vertical plate immersed in a porous medium by using the fifthorder RungeKutta scheme associated with the shooting technique ^{3}. And also the authors Talha Anwar et al. have analyzed the effect of thermal radiation on convective heat transfer across a porous moving vertical plate ^{4}. The authors Nourhan et al. have studied the influence of wall heat flux on the thermal radiation on natural convection fluid flow along a vertical cone immersed in a saturated porous media ^{5}. Recently, many researchers have seemed regarding the interplay of vertical plates immersed in a saturated porous medium with convection flows by solving the numerical techniques. The authors Mahdy et al. and Jha B K et al. have studied the effect of the presence or absence of internal heat generation and thermal radiation ^{6, 7, 8}. The authors Roja P et al. and Jha B K et al. have investigated some related porous medium cases with thermal radiation and also discussed velocity and temperature profiles ^{9, 10}. Our proposed thirdorder nonlinear problem with corresponding boundary conditions is numerically solved by using a newly modified
In recent years, MLWOMM is one of the most widely used wavelet methods for calculating numerical results in differential equations. Nowadays, many researchers have discussed applying numerical techniques to find the results of nonlinear ordinary differential equations (ODEs). In the last few decades, various numerical techniques have been introduced, to name a few, the Shooting method, Finite difference method, Keller box method, Haar wavelet method, etc., these numerical methods are computing the results of boundary value problems (BVPs). MLWOMM is the more effective method for solving the BVPs with finite intervals. Recent research articles are available on various types of wavelets techniques for computing linear and nonlinear differential equations with finite domains, to name a few, Haar wavelet method ^{11, 12}, Daubechies wavelet method ^{13}, Fibonacci wavelet method ^{14}, Chebyshev wavelet method ^{15}, Hermite wavelet method ^{16}, Legendre wavelet method ^{17}, etc. Alfred Haar is the first one who introduced the notation of wavelets in the year 1909 and this work is extended by Grossmann and Morlet ^{18}. The authors Shiralashetti et al. have investigated finding the numerical solution of the strategy of wavelet operational matrix in great detail and used it in differential equations ^{19, 20}. Authors Karkera et al. have briefly investigated the numerical solution of thirdorder boundary layer Magnetohydrodynamics flow due to stretching sheet by applying the Haar wavelet collocation method (HWCM) ^{21}. Furthermore, V. B. Awati et al. discussed the forced convection of thirdorder boundary layer flow problems that are solved in the Haar wavelet technique
The main goal of this article is to investigate the heat transfer problem on the effect of thermal radiation presence in a saturated porous medium. First, to check the efficiency of the MLWOMM solution by implementing it on the boundary value problem and comparing it with the Haar wavelet results with the exact solutions; it gives accurate results compared to the Haar wavelet operational matrix method (HWOMM) results. So we have utilized the MLWOMM to solve the thermal radiation problem and compare it with the velocity or temperature profiles of HWOMM results, and also investigated on effects of thermal radiation with fluid suction/injection. The effects of several selected parameters such as
Wavelets have been employed with great success in a variety of scientific and technical domains. A family of functions is constructed by mother wavelets from dilating and translating themselves into a single function, which we call mother wavelets. The family of continuous wavelets is following ^{14:}
Where,
Where
Where,
The polynomials of the
The satisfy orthogonality criteria for these polynomials are as follows
Here,
In this research article, we assume that
Where,
The
The
Where
The
By truncating the above infinite series, we have
Where,
Eq. (10) represents unknown coefficients of the
where,
and the matrix
By using suitable collocation points are as follows:
For instance, if
In the present section, we construct the operational matrices of
Where,
Therefore, in the following Eq. (15), we obtain the form:
where,
Similarly, we maintain the same procedure, integrating Eq. (15) we will find that
where,
In this section, we developed a numerical solution to a nonlinear thirdorder Boundary value problem and heat transfer problem on natural convection about a vertical plate immersed in a saturated porous medium arising in heat and mass transfer. For the test applicability of the
Following Shiralashetti et al. ^{17}, we assume that
where,
integrating the above Eq. (23) from
again integrating the above Eq. (24) from
In the above Eq. (25) we find
Here,
In this section, we developed a numerical solution for a system of thirdorder nonlinear boundary value problems and the effect of thermal radiation problem on the natural convection of vertical plate embedded in a saturated porous medium, by using the MLWOMM, in the interval
Here,
Now we have substituted the values
If
Solving the above nonlinear system to find the unknowns of








0.0313 
0.0303 
0.0306 
0.0303 
0.0003 
0.0000 
0.0938 
0.0848 
0.0870 
0.0848 
0.0021 
0.0000 
0.1563 
0.1313 
0.1360 
0.1312 
0.0047 
0.0001 
0.2188 
0.1695 
0.1768 
0.1694 
0.0073 
0.0001 
0.2813 
0.1995 
0.2087 
0.1993 
0.0092 
0.0002 
0.3438 
0.2212 
0.2313 
0.2209 
0.0102 
0.0003 
0.4063 
0.2346 
0.2448 
0.2342 
0.0102 
0.0004 
0.4688 
0.2400 
0.2494 
0.2395 
0.0094 
0.0005 
0.5313 
0.2375 
0.2454 
0.2375 
0.0080 
0.0000 
0.5938 
0.2273 
0.2335 
0.2273 
0.0062 
0.0000 
0.6563 
0.2097 
0.2140 
0.2097 
0.0043 
0.0000 
0.7188 
0.1852 
0.1878 
0.1852 
0.0026 
0.0000 
0.7813 
0.1540 
0.1553 
0.1540 
0.0012 
0.0000 
0.8438 
0.1167 
0.1171 
0.1167 
0.0004 
0.0000 
0.9063 
0.0738 
0.0737 
0.0738 
0.0001 
0.0000 
0.9688 
0.0258 
0.0255 
0.0257 
0.0002 
0.0001 
Here,
The associated boundary conditions are:
Here,
Here,
The system of partial differential equations is transformed into a simple nonlinear ordinary differential equation by using similarity transformations. We introduce and use the similarity transformations and parameters ^{2}, we have:
Here,
And Eqns. (30) and (31) become the system of nonlinear ordinary differential equations with boundary conditions as the results follow:
Here, the prime represents differentiation concerning
The speed of injection or suction at the flat plate surface is given by:
Here,
By inserting Eq. (37) into (38), we get the thirdorder nonlinear ordinary differential Eq. (42) connected with the boundary conditions in Eq. (43) we have:
With corresponding boundary conditions are as follows:
We observe the Eqns. (37), (38), and (39) that
here,
For instance 

For 
For 







0.0313 
0.9495 
0.9556 
0.9689 
0.9703 

0.0938 
0.8564 
0.8703 
0.9078 
0.9110 

0.1563 
0.7720 
0.7892 
0.8475 
0.8517 

0.2188 
0.6942 
0.7122 
0.7878 
0.7921 

0.2813 
0.6217 
0.6390 
0.7283 
0.7323 

0.3438 
0.5537 
0.5693 
0.6688 
0.6722 

0.4063 
0.4893 
0.5029 
0.6090 
0.6115 

0.4688 
0.4280 
0.4397 
0.5487 
0.5503 

0.5313 
0.3695 
0.3759 
0.4878 
0.4893 

0.5938 
0.3135 
0.3180 
0.4260 
0.4269 

0.6563 
0.2598 
0.2628 
0.3633 
0.3637 

0.7188 
0.2081 
0.2100 
0.2996 
0.2997 

0.7813 
0.1585 
0.1596 
0.2349 
0.2348 

0.8438 
0.1108 
0.1114 
0.1691 
0.1690 

0.9063 
0.0651 
0.0653 
0.1023 
0.1021 

0.9688 
0.0212 
0.0213 
0.0343 
0.0343 
The study presents that the computations were performed numerically to analyze the boundary value problem represented by Eq. (27) and the heat transfer problem (Eq. (42)) with corresponding boundary conditions provided by Eq. (43) on the semiinfinite domain is solved by using the MLWOMM as described in Section 2.2. First, we solved the test problem of the boundary value problem given in Eq. (27) with the exact solution, the purpose of solving this problem is to check the accuracy of our proposed method and confirms by verifying our MLWOMM results. The collected results are presented in terms of Tables and Graphs. The effectiveness of MLWOMM is presented by comparison with exact solutions and HWOMM solutions which are demonstrated in
Displayed
Overall we observe in the tables and graphs, that the numerical outcomes which are obtained by using MLWOMM are in very excellent agreement with the obtained HWOMM results. On the contrary, the typical Haar wavelet technique produces a good solution, but MLWOMM produces excellent solutions when it is compared with HWOMM solutions. As MLWOMM results are more accurate than HWOMM results. And also vertical velocity or temperature profile rate was studied.
In the present article, we studied the heat transfer of the thermal radiation effect on natural convection around a vertical flat surface immersed in a saturated porous medium. Initially, to ensure the efficiency of the proposed strategy, a thirdorder nonlinear boundary value problem having the exact solution is presented as a test problem. Next, formulated Heat transfer problem, the governing equation of the problem with proper boundary conditions is reduced into the nonlinear ordinary differential equations by applied similarity transformations. The nonlinear problems with finite intervals are numerically solved by using MLWOMM. The effect of the thermal radiation parameters
The authors thank UGCSAP DRSIII for 20162021:F.510/3/DRSIII / 2016(SAPI) Dated: 29th Feb. 2016. And this work is supported by Karnatak University, Dharwad (KUD), Karnataka, India through a University Research Studentship (URS) during the year 20182021: KU.40 (SC/ST) sch/URS/202021/44/544, dated 12/12/2020.