Study of Mathematical Formulation and Numerical Methods for Partial Differential Equations

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6903 Vol. 71 No. 4 (2022)

http://philstat.org.ph

Study of Mathematical Formulation and Numerical Methods for Partial Differential Equations

Mr. Yogesh Tulashidas Khawal Department of Mathematics

Dr. Shoyeb Ali Sayyed

College Name - Malwanchal University, Indore (M.P.)-452016

Article Info

Page Number: 6903 – 6918 Publication Issue:

Vol 71 No. 4 (2022)

Article History

Article Received: 25 March 2022 Revised: 30 April 2022

Accepted: 15 June 2022 Publication: 19 August 2022

Abstract

The numerical answer for circular particles encompassed by substrate arrangement with steady focus is contemplated by Malik and Stefuca. In any case, as far as we could possibly know, till date no broad systematic articulations of substrate fixations in various piece of particles with immobilized compound for round, tube melded, and planar particles have been represented. The examination will be useful in generally exploring enunciation of obsessions for round, planar and tube formed particles by comprehending non-direct differential conditions using He's Homotopy aggravation method. With this investigation, the essential focal point of our own is really introducing an arising meshless method relying upon the possibility of neural organizations for settling differential conditions or perhaps limit worth issues of sort ODE's likewise as PDE's

INTRODUCTION

Linear models of such marvels are approximations of the real world and just every so often these are sensibly precise portrayals. DEs that portray the way of this adjustment quantitatively by modelling each different wonders are by and large halfway and NL. The shut shape arrangement of these DEs regularly emerging in applications can't be acquired although the presence and uniqueness of the arrangement is less demanding to set up. Thus, one is obliged to devise fitting stable numerical strategies for assurance of inexact arrangements utilizing advanced PCs.

Numerical strategies utilize numbers to reproduce scientific procedures, which thus as a rule recreate certifiable circumstances. These techniques have been effectively connected to think about issues in mathematics, designing, software engineering and physical sciences, for example,

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biophysics, air sciences and geosciences. Numerous numerical techniques have been produced to decide arrangements with a given level of exactness.

Relentless state issues are regularly connected with some time-subordinate issue that portrays the dynamic conduct, and the two-point limit esteem issue (BVP) or elliptic condition comes about because of considering the unique situation where the arrangement is enduring in time, and consequently the time-subsidiary terms are equivalent to zero, rearranging the equations.

Two-point NL cutoff regard issues arise typically in different research areas of material science, fluid components, real world models, manufactured energy, e.g., electric circuits examination, dispersing theory, bar speculation, colloidal dissipating, and many body issues, etc., in which the plan of a straight or NL specific customary differential condition is required where out of as far as possible conditions one is embraced at the single point. Incomplete DEs result when any physical circumstance is modeled. These equations emerge in such assorted subjects as meteorology, electromagnetic hypothesis, warm exchange, atomic material science and flexibility, to give some examples.

The numerical arrangement of DEs has been the subject of exceptional movement amid the most recent five decades or somewhere in the vicinity, basically because of advances in PC innovation and the presentation of numerical processing applications like MATLAB, Mathematics, and Maple which thus has prompted upgrades in the numerical strategies that are utilized. Therefore, numerous unmanageable logical and building issues that include linear and NL solitary standard DE and FDE that were already unsolved would now be able to be settled by utilizing fitting numerical strategies.

When dependable and quick electronic means were accessible, the arrangement of an ever- increasing number of complexes DEs ended up plausible. Prior to 1960s, simple reproduction techniques were broadly utilized. As of late anyway these have been supplanted by strategies utilizing complex computerized PCs. various ways to deal with infer appropriate advanced PC algorithms have been produced. The Finite Difference Methods are one method for acquiring estimated answers for conventional or fractional differential equations. Different techniques incorporate Finite Elements, Finite Volumes, Spectral strategies, different spline approximations and so on.

Characterization of second order PDEs

As the PDE emerges from various classes of physical phenomena e.g., consistent thick stream, spread of warmth, reverberation in electric circuits et al., this proposes the administering equations

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are additionally very extraordinary in nature. Characterization of second request PDE’s is an essential idea because the general hypothesis and strategies for solution for the most part apply just to a given class of equations. It administers the number and nature of conditions which must be related with the differential equation all together that a novel solution may exist.

Consider the general second request linear PDE in two independent variables of shape:

𝐴 𝑥, 𝑦 𝜕²𝑢

𝜕𝑥²+ 2𝐵 𝑥, 𝑦 𝜕²𝑢

𝜕𝑥𝜕𝑦+ 𝐶 𝑥, 𝑦 𝜕²𝑢

𝜕𝑦² + 𝐸 𝑥, 𝑦, 𝑢,𝜕𝑢

𝜕𝑥,𝜕𝑢

𝜕𝑦

= 0 We characterize the second order PDE as Hyperbolic, or Elliptic Type likewise as the determinant.

𝐵 𝐶 𝐴

𝐵 is> 0, = 0, < 0, respectively.

The Elliptic class is harmony problems typically depicted as far as a shut area with boundary conditions recommended at all points of the region’s boundary and in this way are likewise alluded to as BVPs.

The Hyperbolic and Parabolic problems are of proliferation compose and ordinarily are characterized on the locales where beginning conditions are given on a piece of the district and boundary conditions on the rest of the part. These can likewise have open-finished locales into which the solution engenders. In numerical speech such complications are known as Initial BVP.

Some usually known precedents of the three kinds are:

𝑢_𝑡 = u_xx(Heat Conduction or Diffusion Equation) 𝐵²− 𝐴𝐶 = 0 Type: Parabolic

Parabolic equations define heat stream and dissemination processes.

𝑢_𝑡𝑡 = 𝑢_𝑥𝑥(Wave Equation)

𝐵²− 𝐴𝐶 = 1 > 0 Type: Hyperbolic

RESEARCH METHODOLOGY

Immobilization of chemical is constraint of catalyst portability accomplished by various methodologies. A few examples are substance or physical immobilization on some surface (particles, plates and so on.), compound or physical immobilization inside molecule, immobilization

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on solvent polymers, cross-connecting of proteins by bi-practical reagents with no transporter and hindrance of catalyst in layer response.

There are two situations when catalyst movement is restricted by dispersion of substrate to compound, for example, response impediment by outside dissemination and response constraint by inner dissemination. The second is talked about in this section.

Catalyst responses partaking in metabolic pathways, guideline forms, or "in vitro" conditions frequently show confounded unique conduct as far as the connection between the response rate and response. It tends to be considered by active NLities coming about because of allosteric collaborations, via autocatalytic systems, or by blend of protein response with mass exchange conditions. This can prompt one of common impacts, the alleged hysteresis, when an impediment of the reaction between response rate and convergence of substrate or different mixes is watched. At the point when the framework is over and over squirmed forward and backward (the fixation is cycled here and there), a hysteresis circle can show up in the response rate focus chart.

It has been recommended that hysteresis impacts in natural frameworks are sufficient to represent transient memory and the presence of hysteresis was depicted just by coupling the catalyst energy and dispersion transport. It was additionally appeared now and again the adjustments in compliance of protein atom may offer ascent to hysteresis. Kernevz and collaborators were built up a mathematical model to study hysteresis circle when mass substrate fixation was continuously expanded and diminished and it was clarified by mix of the substrate inhibitory impact with dispersion constraint in the film. The numerical answer for circular particles encompassed by a substrate arrangement with a steady focus was contemplated by Malik and Stefuca. In any case, as far as we could possibly know, till date no broad systematic articulations of substrate fixations in various piece of particles with immobilized compound for round, tube shaped and planar particles have been accounted for. The motivation behind this section is to determine rough investigative articulation of fixations for round, tube shaped and planar particles by fathoming the non-direct DE subtilizing is Homotopy irritation technique.

A. Direct Methods

Direct strategies such as for instance dimensional analysis method, no cost parameter method separation of variables methods Kline and Abbott (1960) and overall direct method Kruskal and Clarkson do not invoke group invariance. They're probably the simplest and most straight forward methods of determining similarity transformations

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 Dimensional Analysis Method: Dimensional analysis strategy is a technique of minimizing the number and complexity of variables needed to explain a given physical situation by using the info implied by the products of the physical quantities involved. Put simply, it's a strategy for restructuring the original variables measured using dimensions of a situation into a set of dimensionless products (non-dimensional variables) utilizing the constraints imposed upon them by the dimensions of theirs, such non dimensional variables are fewer compared to the originals and might have a far more appropriate interpretation.

 Free Parameter Method: Free parameter technique for determining similarity transformations is pretty straightforward and very simple to use. This strategy is dependent upon the launch of several unknown transformation function(s) of dependent variable occurring in a particular partial differential equation as the product of functions indicated by a possible similarity. One feature of this item is a characteristic of all the independent variables except one. The other functionality is thought to count on a single parameter, say, 3.2. Direct Methods twenty-six where's a variable received from a transformation of variables involving independent variables, particularly which not showing up in the first feature. This technique is utilized also for the troubles with boundary conditions.

Literature Review

Axsater, S. (2015), in their examination paper another strategy mathematical technique dependent on Adomian Decomposition Method has been created. This might be invaluable if adjoining bases are at more limited distances than the focal stop or the outside provider. This paper gives another method to demonstrating such parallel parcels in nonstop survey stock frameworks with one-for-one recharges and Poisson request. We apply this strategy to a two-echelon framework with repairable things.

Alamri, Adel A. &Syntetos, Aris A., (2018), in his exploration paper another decay strategy is introduced to get the express and mathematical arrangements of two dimensional KdV-Burger's condition with beginning conditions. The perception referenced is that the disintegration strategy doesn't need discretization of the factors viz. reality. It's anything but influenced by round of mistakes and need of enormous time and PC memory.

Eissa et al. (2018), in their examination paper acquired lone wave answers for general KdV condition by Adomian Decomposition Method, which has been created by George Adomian. The calculations related with the overall KdV condition examined in this paper were performed by utilizing Mathematika 4. The strategy has been straightforwardly applied without utilizing bilinear

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structures, opposite dissipating technique or Wronskian. The arrangements acquired in the examination can be adequately used to look at the other related issues of logical applications.

Axsater, S. (2016) in his research paper a new approach of two methods is used forsolving the fourth order non-linear. Several exact solutions obtained from each type of non-linearity. The study represents the power of these two methods for solving higher non-linear PDE. The sine method has advantage that it can be implemented in strait forward manner without any need for a transformation for the case where it is not an integer. If the positive equation provides periodic, compactions, solitary patterns and bell shaped solitons then the sine-cosine method works more effective while the solution is kink soliton then tanh method is more effective.

Axsater, S. (2013), in their paper zeroed in on correlation of variational emphasis technique and different strategies utilized for profoundly non-straight conditions. In this paper KdVHirota- Satsuma coupled condition, Kawahara condition and FKdV conditions are settled by variational cycle technique and the outcome was contrasted and different strategies. The perception is that the variational emphasis strategy can settle huge class of non-direct halfway differential conditions adequately, precisely and all the more effectively thus it has been appropriate broadly in material science and designing.

Mehdi et.al. (2017), in this paper presented arrangement of underlying worth issue of explanatory kind, fundamental point of the research work is to discover use of Adomian Decomposition Method for tackling Fokker-Plank Equation and other comparative Des, using methodology in which arrangement of Fokker-Plank Equation is as a focalized arrangement with effectively processed segment. To give the outline of philosophy, they introduced various models in one and two dimensional cases.

Roy, S. S. (2017), gives solutions of coupled Klein-Gordon Schrodinger Equation utilizing the altered deterioration strategy. He considered an arrangement of coupled Klein-Gordon Schrodinger Equations with proper beginning qualities and tracked down its mathematical arrangements by utilizing the adjusted deterioration technique and the arrangement is addressed graphically. This strategy takes out the troubles and gigantic calculation work.

John E. Bernard (2006)The paper introduces a first approximation for a model of understudy execution in tackling linear equations. General qualities of a model are identified with the surviving exploration on equation settling. Wording is denned and used to reconceptualize distributed research results. The model is introduced and is trailed by an examination, as far as the model, of one understudy's endeavors to tackle a specific equation.

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At long last, inquiries for future research are presented which may prompt the explanation of the model. The model relates dimensions of control (i.e., errand, method, technique, process, task, strategy) to the highlights (i.e., remote and close) of equations. The control levels are seen from a data handling point of view; a hierarchal association is proposed with more elevated amounts affecting exercises at lower levels. The model is made to catch the school mathematics way to deal with comprehending linear equations, however it is normal that it will sum up to different settings.

Anthony, Glenda and Burgess, Tim (2014) Concerted endeavors at enhancing understudy execution in variable based math show that "youngsters all through the rudimentary evaluations are equipped for adapting amazing bringing together thoughts of mathematics that are the establishment of both number juggling and polynomial math". In New Zealand, Britt and Irwin's (2005) examination of the Numeracy Development Project found that those understudies who procured adaptability in utilizing a scope of general arithmetical methodologies likewise built up the capacity to express the structure of those techniques in representative structures.

Aisha et al. (2007)In this investigation, various variables influencing understudies' differential conditions (DEs) settling capacities were investigated at pre college level. To investigate primary variables influencing understudies' differential conditions critical thinking capacity, articles for 19 - year time frame, were basically surveyed and examined, uncovered that blend of 4 elements;

"epistemological numerical statement tackling convictions, conviction about value of science, self- managed learning (SRL) systems and objective directions" can possibly improve differential condition issue capacity.

In light of these discoveries, a theoretical model is created and introduced to upgrade DE critical thinking capacity. This investigation has given a few significant ramifications to the educational program planners and educators to improve theoretical comprehension in differential condition critical thinking, especially in the agricultural nations.

Mondal et al. (2016)The numerical computation for handling "first-demand direct differential condition in feathery environment" is discussed. An arrangement, specifically, "Runge-Kutta- Fehlberg strategy," is depicted thoroughly for tending to the said DE. The numerical plans are differentiated and (i) - gH and (ii) - gH differential (exact courses of action thoughts) structure. The approach follows total blunder appraisal. The combination of Runge-Kutta-Fehlberg strategy has been talked about. The interaction strategy is applied to a mechanical issue in fluffy climate which shows that it is a promising technique to address the said kinds of respectful condition. Later on we

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can apply these techniques for tackling higher request direct and nonlinear differential condition in fluffy climate

MATHEMATICAL SIMULATION

The dispersion conditions for the limit conditions are disentangled by numerical strategies. The capacity PSEs in Sci-lab programming which is a component of dealing with as far as possible regard issues for incomplete differential condition was used to understand these conditions. The numerical results are also differentiated and our efficient results in Tables 1for round and barrel shaped anode. In every one of the cases the typical relative bungle among orderly and numerical result is under 3%.

Table 1: Examination of standardized substrate focus c for different upsides of β when m=800, 𝚽 = 𝟐𝟎 (spherical particle)

X 𝛽 = 20 𝛽 = 50 𝛽 = 100

Eq.

(3.10)

Numerica l

% Eq.

(3.10)

Numerica l

% Eq.

(3.10)

Numerica l

%

Deviatio n of eq.

(3.10)

Deviatio n of eq.

(3.10)

Deviatio n of eq.

(3.10) 0 792.1

5

796.00 0.4860 785.1 8

792.40 0.9195 798.3 5

799.20 0.1064

0.

2

792.4 7

796.10 0.4454 787.7 7

792.70 0.8819 798.4 1

799.20 0.0989

0.

4

793.4 1

796.70 0.4146 787.5 5

793.70 0.7809 798.6 1

799.30 0.0864

0.

6

794.9 8

797.50 0.3169 790.5 1

795.20 0.5932 798.9 4

799.50 0.0700

0.

8

797.1 7

798.60 0.1793 797.3 0

797.30 0.3322 799.4 0

799.70 0.0375

1 800 800 0 800 800 0 800 800 0

Average deviation 0.3070 Average deviation 0.5846 Average deviation 0.0665

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Table 2: Correlation of standardized substrate fixation c for different upsides of β when m=800, Φ=20 (round and hollow molecule)

X 𝛽 = 20 𝛽 = 50 𝛽 = 100

Eq.

(3.10)

Numerical % Eq.

(3.10)

Numerical % Eq.

(3.10)

Numerical %

Deviation of eq.

(3.10)

Deviation of eq.

(3.10)

Deviation of eq.

(3.10) 0 792.15 798.30 0.1106 788.23 796.40 0.9857 777.78 7992.40 1.8797 0.2 795.31 798.40 0.0993 788.70 796.10 0.9832 778.66 792.70 1.8031 0.4 795.90 798.60 0.1005 790.11 796.70 0.8340 791.33 793.730 1.5832 0.6 796.87 799.00 0.0878 792.47 797.50 0.6347 785.77 795.20 1.2001 0.8 798.24 799.40 0.0501 795.76 798.60 0.3568 792.00 797.30 0.6691

1 800 800 0 800 800 0 800 800 0

Average deviation 0.0747 Average deviation 0.6249 Average deviation 1.1892

the new and straightforward inexact scientific articulations of centralizations of substrate c (m, Φ=20) for circular, tube shaped and planar molecule individually. The upsides of Thiele modulus % were in the reach 20-400 with standard worth 160 , dynamic boundary and were in the reach 10-300 with standard worth 128 and substrate fixation on the molecule surface m were in the reach 100- 1000 (Malik and Stefuca 2002). The convergence of substrate c depends on the three boundaries Φ, and m. The Thiele modulus Φ may be adjusted by varying thickness of synthetic layer or fraction of compound immobilized in organization. In the compound layer, this border depicts the overall meaning of dissemination and response. When the Thiele modulus Φ is nearly zero, the energy is the transcendent resistance, and the entire substrate take-up is actively regulated. The substrate focus profile is virtually homogeneous under these conditions. The total amount of dynamic compound directs the whole energy. At the point when Φ is gigantic, dissemination restrictions are the head choosing variable.

x Φ=20 Φ=100 Φ=150 Φ=200 Φ=250 Φ=300

0 799 710 610 490 290 20

0.2 799 711 611 500 300 80

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0.4 799 720 620 550 350 150

0.6 800 750 680 600 400 200

0.8 800 770 700 700 600 600

1.0 800 800 800 800 800 800

(a)

x 𝛽=10 𝛽=60 𝛽=120 𝛽=200 𝛽=300

0 790 740 690 630 560

0.2 792 750 700 640 580

0.4 794 760 720 650 600

0.6 796 780 740 690 650

0.8 798 790 780 750 700

1.0 800 800 800 800 800

(b)

Fig. 1 Plot of standardized focus profiles for substrate c versus the standardized distance x for the round molecule The fixations are registered utilizing Eq. (4.10) for (a) different

0 100 200 300 400 500 600 700 800 900

0 0.2 0.4 0.6 0.8 1

0 200 400 600 800 1000

0 0.2 0.4 0.6 0.8 1

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upsides of β and some fixed upsides of m and Φ m=800, Φ=20) (b) different upsides of β and some fixed worth of Φ and m (m=800, 𝚽 = 𝟐𝟎).

Plot of standardized focus profiles for substrate c versus the standardized distance x for the round molecule, the fixations are registered utilizing (a) different upsides of β and some fixed upsides of m and Φ m=800, Φ=20) (b) different upsides of β and some fixed worth of Φ and)).

x Φ=20 Φ=100 Φ=150 Φ=200 Φ=250

0 790 690 610 310 20

0.2 790 700 620 350 50

0.4 790 720 680 400 150

0.6 800 750 700 600 200

0.8 800 790 780 700 600

1.0 800 800 800 800 800

(a)

x 𝛽=10 𝛽=60 𝛽=120 𝛽=200 𝛽=300

0 790 710 640 650 460

0.2 790 730 670 660 480

0.4 792 750 700 670 500

0.6 796 770 750 690 550

0.8 798 780 770 700 650

1.0 800 800 800 800 800

0 200 400 600 800 1000

0 0.2 0.4 0.6 0.8 1

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http://philstat.org.ph (b)

Fig. 2. Plot of standardized focus profiles for substrate c versus the standardized distance x for the round and hollow molecule The fixations are figured utilizing different upsides of Φ and some fixed upsides of m and β (m 800, β 20) (b) different upsides of and some fixed worth of Φ and m (Φ 50, m 800)

Address both round and tube shaped molecule substrate fixation c for various upsides of dimensionless dynamic parameter β. From these figures it is deduced that the grouping of the substrate c increments when active boundary Ki/Km diminishes. The standardized three- dimensional substrate fixation profiles for circular, tube shaped and planar molecule are plotted in Figwhere the information given by past figures are checked. The dimensionless substrate fixations for the circular, tube shaped and planar molecule are plotted in Fig From these Figures it is construed that the dimensionless substrate fixation for circular molecule is more prominent than planar and round and hollow molecule.

0 100 200 300 400 500 600 700 800 900

0 0.2 0.4 0.6 0.8 1

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Fig. 3 The three-dimensional standardized substrate fixations c for (a) round molecule when 0≤φ≤50 β=10 (b) tube shaped molecule when0≤β≤50, Φ =50 (c) planar molecule when0≤m≤300, β=50.

x φ=50 (spherical) β=300(cylindrical) m=800(planar)

0 590 420 0

0.2 595 450 0

0.4 600 500 0

0.6 650 550 0

0.8 700 600 0

1 800 800 800

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Fig. 4 Plot of the two-dimensional case outline of the standardized substrate focus c versus the standardized distance x on account of round, tube shaped, and planar particles The standardized substrate fixations were processed utilizing Eqns. and when the parameters, φ=50, m=800, β=300.

A numerical model for an immobilized protein anode has been depicted. In this section the molecule balance condition have been planned and addressed under consistent state conditions subject to characterized limit conditions. In this section, we got a scientific articulation of the standardized substrate focus for all potential upsides of dimensionless motor boundaries and the Thiele modulus % for round, tube shaped, and planar molecule. The basic shut types of insightful arrangements have been proposed utilizing Homotopy bother strategy. Besides, it gives great concurrence with recreation obtains.

Conclusion

As Homotopy perturbation technique is greatly available to non-mathematicians and engineers. It has proposed HPM, this technique has been considered an incredible scientific tool for different kind of non-linear problems, as it is a promising and advancing technique. In addition to its scientific significance and association with other branches of mathematics, it is broadly utilized in all the ramifications of current science.

Different perturbation methods have been utilized to handle non-linear issues. Sadly, the conventional perturbation methods rely upon the doubt that small parameter should be present, which is over-exacting, making it hard to find more extensive applications because most non-LEs have no little parameter using any means. Therefore, numerous new techniques have as of late acquainted some ways to wipe out of the small parameter like artificial parameter method, which is introduced, Liao proposed HAM, VIM proposed, differential transform method (DTM) by Zhou,

0 200 400 600 800 1000

0 0.2 0.4 0.6 0.8 1

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ADM by Adomian and others. Now we discussed the various authors those who have used these semi-analytical techniques on different classes of physical problems

It has proposed a solution for non-linear stochastic equations by considering different types of non- linearity. Further, Adomian has applied the decomposition and asymptotic decomposition method for the different non-linear and the system of non-linear PDEs. Adomian applied decomposition technique on various physical problems like duffing equation, non-linear transport equation, matrix Riccati equation, advection-diffusion and dissipative wave equation in which the non-linear terms are handled with Adomian’s polynomial, and they conclude that this method gives a viable technique to provide the precise solution of a wide class of dynamical system which represents the real physical problems. Adomian provides the solution of coupled non-linear PDE with uncoupled boundary conditions.

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