 I just want to give an introduction to the mathematical modeling of water resource system here. So, I know most of you have gone through the courses like numerical methods. So, I will have a brief discussion about the mathematical model, what are the essential steps in mathematical modeling and then we will discuss to important numerical methods, finite difference and finite terminal method and its applications. So, here in this first topic the mathematical modeling water resource system, what is the model and how, what are the essential steps in the modeling framework and then what are the methodologies available for modeling. So, here as I mentioned if it is a watershed or surface water or lake or if it is an aquifer system in all these cases we want to see what is going on. So, as I mentioned we can integrate various systems like soft computing techniques N and GA or we can integrate numerical methods with respect to GIS geographic information system which. So, everywhere all these various computing techniques and soft computing techniques or numerical methods or geographic information system or remote sensing all these things we can integrate together. So, that we can have better models, we can have better development and then our aim is to see the variables like either if it is aquifer parameters and aquifer variables means it is head and concentration if it is transport modeling or if it is the flow is concerned the lake hydrodynamics or hydrodynamics in a river or x-rays all these problems we can develop numerical methods and then we can identify what is happening. So, various methodologies we can we can combine together and we can have integrated models. So, as I mentioned the motivation for modeling is mainly to solve the problems of practical interests with a large number of complex variables. As I mentioned say if you deal with an aquifer system very heterogeneity will be the major issue then anisotropy then the porosity will be varying, hydroconductivity will be varying all these things we have to take into account while modeling. So, we are trying to represent the real system in a simplified way so that we can have some reasonable way of prediction we may not be able to completely replicate the real field system with respect to computational models, but the possible way we are trying to do that and then we are trying to develop a model. So, as I mentioned if it is a groundwater pump groundwater for water supply to remove pollutants from the unsaturated zones to reduce contamination of groundwater and say lake models or watershed modeling like that various kinds of models we can develop and then this models provide information required by say managers whether it can be aquifer management or it is lake management reservoir management all this management we need some way or another way of modeling so that it will give the various conditions, various scenarios if this is this much what is pumped out of the aquifer system what will happen or if the reservoir is operated this way what will be the scenarios. So, we are generating various scenarios and then we will as engineers and scientists we will be generating the scenarios and then giving to the decision makers and then decision makers will decide which way they have to go. So, in all this aspect you can see that essentially what we are doing is one way or another way of management that means say whatever we are doing through modeling process we are helping the managers to take a managerial decision so that a better prediction, better way of say implementation can be done for the considered problem. So, we need models to solve a managerial problem so it is helping for planning and operation of the considered system in order to achieve certain goals without violating technical and non-technical constraints. So, as the water source system which Professor Jyothi Prakash would have already discussed obviously for any kind of problem say if there will be certain constraints it may be legal constraints it may be technical constraints different kinds of constraints will be there and then we will be having an objective that means here say if you are considering a road water system our objective is to see that we want to pump as much of water as possible from the aquifer system without filing the head certain level that means say there are certain constraints so that it will not affect the aquifer system and then it will be replenished annually with respect to the rainfall. So, within the certain constraints we want to see how the system is behaving so that is the analogy of any kind of modeling so there will be an objective function there will be there will be a number of constraints so within the constraints we want to get the best so that is what we are doing. So, say for example the case of subsurface remediation contamination caused by a spill of chemical products as we have already seen in that the animations so the decision what that type of decision we want to take what kind of remedial activity should be selected where it should be the remedial activities should be undertaken and what level of activity should be involved so like that a number of questions will be there so through modeling what we are trying to do we are trying to answer these questions in an efficient way so that is what we are trying to do in any kind of model. So, a model may be defined as a selected simplified version of a year system which approximately simulates the lattice excitation response relations that are of interest so we want to see we have certain conditions with respect to that condition how the system is going to behave that is what we are trying to do. So, in any kind of modeling we will be selecting a typical domain and then with respect to that domain for the given field condition we will be putting certain assumptions and we will simplify it and there will be a number of approximations will be there and then they will give certain excitation to the system and then how the system will be behaving so that is what we are doing in the modeling. So, it is finally what we are getting is excitation response relations so that is what we are finally achieving through any kind of modeling. So, all these details are there in the lecture notes so what we can do is we are trying to predict the behavior of the considered system and then we will be getting through modeling a better understanding of the system and that will comply with the regulations. As I mentioned for any kind of water resource system is considered is it a reservoir or is it a watershed or if it is groundwater basin anywhere there will be we have to comply certain regulations. So, to meet those regulations so we will be doing modeling and we will be generating various scenarios and then finally we can choose the best scenario. So, in this way we can see that this models gives always information that is very key to the design of the particular system and then that will be that can be used for the implementation in the field. So, if you consider a watershed like this you can see that watershed is considered we will be having the rainfall. So, rainfall is one of the excitation to the system and then if you are going to construct a reservoir at particular location so annual basis on an annual basis we can calculate say how much will be the runoff taking place and how much storage we can have at any particular location of the for the river which we consider. So, that kind of the way with the modeling which we want to do. So, a model is there will be always an excitation as I mentioned for the watershed is considered we can have the rainfall is the excitation and response is how much is a runoff at particular locations. So, in any kind of modeling there will be the domain will be we will be defining a domain and then within that domain there will be certain conditions and with respect to that conditions we will be giving certain excitations. So, then we want to see the response. So, similar way if it is a groundwater aquifer system then we will be we will know the we can demarcate or delineate the aquifer system and then we will be having the boundary conditions like we will be specifying what is at various boundaries and then we will be seeing if there is a pumping well how the head is varying. So, that is the response. So, any kind of model there will be an excitation and then what is the response. So, that is what we are trying to model. So, as I mentioned there is a lab role for the modeling in any decision making process. So, we say as a consultant or as an engineer or as a scientist you are having the problem and the client will be coming with say they have certain objectives. So, that objectives are given to you and then according to that objectives you collect the various data and then you can identify what are the possible alternatives solutions and then with respect to that we go for various kinds of predictions and then we will be evaluating the objective function. As I mentioned for any kind of model there will be an objective function and then there will be constraints. So, we can evaluate the objective function and then we can select the preferred alternative and then the decision makers will try to implement that alternative and of course, this modeling generally any kind of especially water source system is considered it is a continuous process. So, once we develop the model you cannot just you have to see that you have to continuously monitor whether the system is behaving properly as we expect. So, that means, say whether what what we are looking for whether it is happening. So, that way you have to monitor the system. So, within this framework we can define the modeling process as so, we need a certain data. So, the required information with respect to what are your objectives accordingly you are having you will be looking for the information. So, you may collect the data through field work or from various other sources and then you first conceptualize the model there is this is the most important part of any kind of model if it is system model or physical based model which we are going to discuss here any type of problem the conceptualization and then to develop a mathematical model. So, conceptualization is the part where as an engineer or as a scientist you have to put all your efforts with the conceptual modeling is right then all other things will be right, but if the conceptualization itself is wrong then your model will not behave the way which you want. So, once it is conceptualized the next step is we can we will be trying to if you are going for numerical or mathematical modeling you will be trying to represent the the conceptual model with respect to governing equations with respect to boundary conditions with respect to initial conditions etc. And then once the conceptualization and mathematical model we have already selected then we we go for what kind of solution methodology you want you want to simulate whether something in the laboratory or whether you want to develop a computer model and then whether it is analytical way of solving or numerical way of solving which way you want. So, we can choose it and if a software is available you can choose that's a particular software or you can develop your own computer code or numerical model and then you can go for predictive simulations and then as I mentioned in this there will be say most of the time we are trying to simplify your real system a real field case with respect to various assumptions. So, there will be number of uncertainties will be their sensitivity for each parameter we have to do and then we have to come back and again go for predictive simulations and also the one some of the other aspects are calibration that means you are trying to develop a model for a given field problem. So, you have to calibrate various parameters with respect to the behavior of the model and then of course whether your model is giving what it should give. So, you have to validate the model and then you have to finally again come back and do the predictive simulations. So, like this so here are various steps we will have a brief outlook with respect to the flowchart. So, in any kind of modeling the procedure starts the step number one we add in the file the information required for management decisions. So, as I mentioned we are say as engineer or as a scientist say a client will be coming to you or your manager will give a problem to you. So, that means the objectives are known to you. So, this way they want to say for example, if it is say groundwater well is to be established and then the owner wants to see how much water can be pumped without creating any problem to the aquifer system or if a state government is planning to construct a dam and a reservoir then they want to see that say for the given rainfall how much water can be stored for the at a particular location where the dam is going to construct and the reservoir is there. So, how much is can be the storage for the annual for the given annual rainfall conditions. So, the objectives are known. So, once the objectives are given to you then next step is you can identify what kind of information are required to solve the problem. So, say for example, the groundwater flow model as I mentioned the various things here the water levels at selected points spring discharge boundary discharge or recharge concentration specified points geological parameters hydraulic parameters etcetera. So, you can identify what are the parameters or data needed and then you can say that this is the first step in any kind of model. So, based upon the objectives what is given to you and then you can visit the field where we have say the modeling is going to be done if it is required and then you can define the domain of the problem and then next step is development of a constructed model as I mentioned. So, here the process is that the real problem is very complex. So, the real the complex problem we are trying to simplify. So, to create to make a mathematical model. So, in this process a number of assumptions we have to put and then you have to conceptualize the problem. So, the conceptualization whether you want to develop a three dimensional model or you want to have a two dimensional model or a one dimensional model depending upon what kind of say things you want. Say for example, when you are doing river model. So, it is with respect to the flow of the river we want to see how the velocity is changing or with respect to the head is changing. So, in that case a one dimensional model may be sufficient or if you are doing an aquifer model most of the time a two dimensional model may be sufficient. So, depending upon what kind of the way you want and your resources you can identify you can simplify the system and then you can conceptualize the system. So, in the conceptualization the important parameters which you have to consider the geometry of the domain. So, that is very important you have to delineate whether are aquifer system or watershed or with the area where we are going to model. So, the geometry is very important and then the effect of various parameters if it is aquifer system it can be heterogeneity or anisotropy or if it is watershed then how are the land use land cover and then various other parameters we have to see and then with respect to all these parameters you will be conceptualizing the model. So, the conceptualization as I mentioned it is the most important part. So, here be the objectives of the investigation according to you only we will be conceptualizing the problem and then what are the available resources. So, when you are going to do if you want to see the aquifer head variation if you want to do a three dimensional model you may need a very high power computer or even sometimes three super computer may be required to model in large aquifer system. So, instead of going for such a complex model even if you develop a two dimensional model that may give the the results which you want to certain level of accuracy. So, that way the depending upon your the available resources you can simplify the problem and then depending upon the data availability. So, you can see that especially this subsurface or groundwater systems are considered you can see that most of the time the data is a major problem. So, you have to put your you have to limit your modeling in such a way that with respect to the available data or whatever data you can collect accordingly the modeling should be done and then of course, the legal and regulatory framework which pertains to the considered case. So, that is what the there will be technical constraints there will be legal constraints there will be non-technical constraints or number of constraints will be there. So, you have to conceptualize your problem accordingly this so that we will be having the best model. So, the conceptualizes is the most important part and then as I mentioned once the conceptualizes is done then almost 50 percent of the work is over since the accordingly if the conceptualizes is done properly then you can give the work even to your assistant so that other things are much easier. So, once the the conceptualizes is done in step number two, step number three development of a mathematical model. So, here in this step the conceptualization model is expressed in the form of mathematical model which may consist the definition of the geometry of the surfaces whether it is rectangular type surface or is it a three dimensional model or is it irregular boundaries then equations that express the balances. So, we can see that the hydrodynamics is concerned in lakes or the rivers we may use the Navier-Stokes equations or we may use St. Wiener's equations or the ground water is considered we may use the flow equations and transport equations. So, the equations that express the balances and then flux equation that relate with the flux that means how much is the inflow coming to the system how much is outflow going from the the considered system. And then if there is sources things like a pumping then recharge then of course, if you are going for a time cn model time dependent model of course, the most important we have to start with something that means when time is 0 you have to see what is the head variation concentration variation that is the same like the case of hydrodynamics in rivers or lakes also we have to see what is the water level conditions that is so called initial conditions. And then of course, the model is behaving according to the given boundary conditions is the initial condition is when for a time cn model the starting we give some conditions that is so called initial conditions but boundary conditions if how accurately you represent the boundary conditions accordingly the model may behave in a better way. So, the boundary conditions are very important in any kind of model. So, this is the third step where we develop a mathematical model. So, then once the mathematical models are developed now you can take a decision what kind of whether for the given mathematical model whether there is an exact solution that means mathematically analytically you can solve it and then you can find out a solution that is may be available in literature. So, you can choose if it is available but most of the real field problems are considered the analytical solution will not be possible to apply. So, you have to go for some approximation for the mathematical equations which we have already developed or which we have already obtained in step number 3. So, here the we may use the numerical methods like finite difference finite element or various a number of methodologies are available in literature. So, you can choose say this depends upon your familiarity if you know finite difference method then you may go for finite difference method but the accuracy may slightly vary but depending upon your familiarity how much time is given to solve the problem and then what kind of software is available accordingly you can choose the numerical method which we are looking for. So, as I mentioned here most of the time some software may be available it will be better to purchase the software directly and solve it or you can develop your own codes but only when you develop a code of course you have to verify how your model is behaving. So, next step say even if you buy a software also most of the time it will be very good to verify the code what is whether what we are looking is the software which we develop or which we buy it will be having say its own objectives it can do say what kind of simulation all these things will be given. So, we want to see that whether it is giving what it should give. So, verification involves comparing solutions obtained by using the code with those obtained by analytical methods. So, if any analytical solution is available for a very simple domain you can apply the code and then see that with respect to that analytical solution whether you are getting what it we should get. So, whether how much accurately we can get the solution and then if any lab models or laboratory models are there that also you can try to simulate and see that whether the code is giving what it should give. And then next step is so called model verification is not to be essential that with respect to your field problem but model validation next step number 6 is so called a model validation. So, the model validation is now we are directly coming to the real objectives of your work and with respect to your objectives. So, if say for example, for an aquifer system if you are trying to model and if any offset values are available then you can just check with respect to the offset values you can develop the model and then you can run it and see that whether you are getting very near to that value what is available with respect to field observation or with respect to laboratory if any model is available. So, a model validation can be with respect to laboratory experiment very similar way which we develop for the given field problem or if any field values are available we can use that. So, that is step number 6. And then as I mentioned since these are all very not so complicated things that is why I am going slightly faster. So, the step number 7 we can see that when we are dealing with any kind of real problem real field problems are considered it is so complex we will be assuming a number of parameters we may assume very many of the variables with respect to given system. So, like an aquifer system is considered we may see that the hydraulic conductivity will be varying actually for a heterogeneous system it will be varying from one point to another point. So, you cannot do all this kinds of say modeling with respect to continuous variation. So, what you do you may see that the k value for this particular zone is 40 meter per day another zone is 50 meter per day. So, like that you may be doing a sonation and then you will be putting the parameters. So, very similar way the porosity is considered we may be giving some parameters and then say if you are dealing with the hydrodynamics. So, then what is the diffusivity what is the values of the various coefficients then we may be putting certain with respect to the field condition what is available with respect to literature or with respect to some field measurement we will be putting to the model. So, but actually to see that what you are putting is right with respect to total behavior of the model you have to calibrate the model and then you have to see how the parameters is working properly or not. So, the step number 7 we calibrate. So, the calibration is generally done with respect to say for example, for the given problem if some field values are available then you can assume certain parameters say to the reasonable range and then you can run the model and see that what is with respect to the observed values whether you are getting the same very similar way to certain level of accuracy. So, then if it is not getting you can slightly again change the parameters and then see how the system is behaving. Nowadays say actually the so called soft computing techniques like ANN, the genetic algorithm etcetera we can use for inverse modeling. So, inverse modeling is that we can combine the so called numerical methods with respect to the artificial neural network and the actually you know that artificial neural network type soft computing tools are just like a black box it is not but we are discussing here is very much based upon the physics of the problem. But the black box model like attribute neural network what it is doing is say we are having some input some output. So, it is trying to generate some correlation between the input and output and then just trying to predict. So, but in the real physically based model we are trying to really simulate what is happening within the field even the modeling is very complex compared to the just use of the black box model. But if we can combine together. So, nowadays what we are doing most of the time we may combine these kinds of numerical models with respect to a soft computing tool like ANN, genetic algorithm etcetera. So, that wherever these parameters are there model the parameter estimation is considered we can do it through inverse modeling we can find out the real parameters with respect to some of the offset values. So, if we can combine the numerical models with respect to the soft computing tools it was found that very very good where your prediction is possible. So, step number 7 we do model calibration and we do some parameter estimation. Then so, also most of the time we may go for you can see that say each of the parameter it will be with respect to the parameter variation there will be a lot of variation with respect to the given variable say for example, the head is considered say if the k value the hydraulic conductivity is 2 percent varying 5 percent varying 10 percent varying. So, then you can see that head also will be varying accordingly or the velocity also changing accordingly. So, you can do a sensitivity how sensitive is your model for the given condition. So, you can do this through a sensitivity analysis. So, the sensitivity analysis is also very essential when you are especially developing a model. So, once you do this 7 steps now your model is ready for application. So, application means now what your client want your client want to see say for an for example, for an aquifer system this much is pumping now how the head will be varying in the aquifer system or if a contaminant plume variation then if this much is pumping out or this is the condition you they want to see how the scenario or like hydrodynamics or river modeling everywhere. So, for the given condition they want to see the client want to see what is your condition. So, now the first the 7 steps which we have discussed is to make sure that you are having a system or a modeling system where you can apply directly and get solutions. So, in step number 8 now you are doing the problem the real problem and now you will be getting the output and then you can just simulate various scenarios for the given conditions and then you can get the results. And then in step number 9 generally you can see that most of time a number of uncertainties will be there or stochasticity that means say for example, we are dealing with reservoir operation or we are dealing with the watershed modeling you can see that even though we say that animal rainfall is known, but you can see that lot of uncertainty will be there probability it is very much say even if we assume that it is 115 centimetres the animal average rainfall in India, but it may vary from one location to another sometimes it may be more sometimes it may be less. So, we have to see the uncertainties what are the uncertainties what are the probability of the various given conditions. So, uncertainty analysis is very important in this kinds of modeling. So, this is about many elements associated with the model such as is selected construction model appropriate or the values various coefficients used are correct or the selected boundaries are appropriate or the excitation that is mean as I mentioned the rainfall is this much is possible or it can totally vary. So, all these parameters we can do an uncertainty analysis and if in case if it is needed we can go for stochastic modeling also. So, stochastic models the information coefficient introduced say what is the probability distribution what are the possibilities. So, accordingly the system may change. So, that also we can do in the type of model which way we may go. So, now once all these nine steps are done now you can come with now you have already simulated various scenarios. So, now you can summarize yes if this is the condition these are the output these are the results. So, now you can analyze your results then you can graphically represent the results and then you can put certain conclusions that means for the this given scenario one these are the conclusion that means how the hairdo will be varying how the concentration may be changing how the lake hairdo dynamics will behave or how the reservoir operation can be controlled. So, all these things will be coming as conclusion for the given conditions and then even next step is model reporting. So, I hope all of you know what will be there in the model report. So, you may start all these what we are discussed there will be definitely should be there in the report. So, you may how the abstract and then introduction modeling objectives conceptual model, mathematical model, model coefficients and parameters, code selection or development, model calibration, parameter estimation, model runs sensitivity analysis stochastic analysis and then your own predictions and then results and discussion. So, this is a typical model report just like what we will be doing. So, know all these things you can see we are trying to simulate a real system. So, do not think that whatever is giving completely correct that is most of the time it is not right. So, you should apply your mind in such a way that the model uses concerns you can see that by running the model what is to be expected in the future we are getting an observation and early warning network can be designed. So, that is the purpose, but most of the time there will be we cannot always say it is 100 percent right or 90 even say that probably the conditions we have to always tell the clients. So, otherwise the clients may think that they have given this and then that should be completely represented. You can see that most of the time the clients coming to you they may not know many of these aspects. So, you have to specifically say the model use, model applications, how much reliable the model is, how much sensitive various parameters and how you have put this model in this framework. So, that the system may behave accordingly. So, it is all these aspect model use, model applications, the reliability of the model, then the limitations of the models, the advantages of the models all these things you may have to specifically say in your report. So, that say it will be clear. Nowadays you can see that say you cannot hide many of the things now in India also earlier in western countries it was in India also the right to information act is very much there. So, you have to see that whatever you are putting it should be very clear you have to write everything clearly otherwise what happens that say the clients may go and see you in the court. So, that you will be in trouble say for example recently we did some work for the New Bombay Municipal Corporation and then what has happened is that it was actually so you know that while dealing with the contracts there will be a number the the contractors will be trying for a particular work and then there will be contractors rivalry. So, what has happened is Siddharth has given the New Bombay Corporation has given some work to particular contractor, but we were not involved IIT was not involved, but we were only checked their proof that means whatever the report is right or not. Then what has happened is opposite the contractor who could not get the work he went to write RTI you know RTI is very powerful now. So, now the RTI came and told that okay IIT has to give ender what is given for whether we want to see what you are done is right actually there was nothing we are done completely correct everything is right, but the driver contractor they want to see that this work is not say since they could not get the work. So, they want to see they want to put the municipal engineers in soup and the opposite contractors also in soup. Then what has happened we have to show A to Z to the RTI and then what happens you can see that most of the time we are all in another soup. The problem is that we are having a contract obligation with respect to the clients that means the other contractor who has mentioned that whatever they are given we cannot disclose to anybody. So, we are in a soup since as per RTI right information act we are supposed to give everything, but from the other side there is legal complications that if you give this thing without their consent then we will be we have we will be implicit. So, these kinds of issues will be coming. So, what we need in any kind of reporting you should be very careful since now all the people are very sensitive they know what you what way they can get the things. So, most of the time there will be rivalries between the contractors or the companies. So, this also you have to as a consultant you have to deal all this aspect. So, just I told a small example we are in a soup last month regarding some work we did for a municipal corporation. So, anyway we will come back to our modeling. So, now we have seen the total framework of the model starting from our objectives then to the reporting. So, now my purpose of this lecture today and the afternoon is about the the modeling techniques. So, the as I mentioned where I say now once you develop a conceptual model and then we are having a mathematical model now next step is what way you want to solve the problem since now once you develop the mathematical model that to that is only the first step for the solution. So, the solution is concerned as I mentioned there are number of methodologies available. So, the first one you can as a say you can see that most of the time we are engineers are very much mathematical oriented. So, once you get the equations you will be looking to the equations and then see that whether there can be an exact solution available for that equations. So, that is where we can use the so the method called analytical or exact solution. So, you can see that most of the problems since you can see nowadays and not now last 30 or 40 years people are trying to do PhDs, master's all this we want to do something to show that you are doing something all the time that everybody knows. So, all the problems most of the problems are already solved let us let me tell frankly. So, that is why we are trying always to couple something with another so that another creation will be done and you may get PhD or you may get a master's thesis or whatever it is. So, all the time most of the works are done and if you begin the literature your most of your problems can be if you are working as a consultant and I see most of your problems are already solved and you can only thing is that you have to take something from somewhere and knit it together and then your solution will be there. But say let for to see that you should understand you should have broad knowledge. So, most of the time when I teach master's and PhD students I tell that even though it is you may think that why the professor is giving so much of broad based knowledge it is not required but see that if the things are available if you know if you immediately to click on your mind you know need to study in detail but if the things are available then immediately you can go and refer and get the things done in a very fast mode. So, that is what the you should know all these things at least in a broader way. So, the solution methodologies are concerned the first one is analytical method and then second one is so called a physical method. So, physical method you can see that before this computational methods we are coming to picture say by 1950s or 60s after the first computer has come and then then there was a revolution in that area. But before that all the time people say if you want to see how an aquifer is behaving with respect to pumping what you have to you have to construct in to certain scale and then there will be Reynolds law, Froude law there will be a number of similitudes we have to apply and then you have to construct and then see what is happening. So, all of you would at least some of the in your laboratories there will be some or another way of physical model will be there. So, with respect to that so, you can simulate say the given conditions and say and then you can find out what is what will happen realistically. So, even though it may not truly represent since you are there also you have to put a number of approximations. So, even though it may not truly represent what it should give. So, physical method is very much used for a long time maybe more than hundreds of years and then also still also it is very much used actually last maybe 99 during that time. So, what has happened is you can see now in most of laboratories even including IIT. So, most of the laboratories are in a very very bad conditions. So, what has happened is in 80s and 90s when these computers came into picture then people thought that we do not need any kind of laboratory it is everything we can do in simulation through the modeling, but what is the reality is that it is not right. So, many of the things say there is limitation for the computational models. So, we cannot completely replace a physical models with respect to computational model that is why now there is a we say a revived interest in physical models and in most of the good universities abroad as well as now also we are say we are making good laboratories, we are in the process of making good laboratories here in IIT also. So, physical method has got its own advantages and then in the overall way it will give a better better way of visualization than the any kind of computational methods. So, the second one is a physical method and then computational methods or both this can combine together most of the numerical methods we cannot solve without computer. So, computational methods what we are trying to do we are handling the given equations we are having the domain. So, with respect to this all these we are trying to develop a computer code and then we will be writing the code and then we will be applying the boundary condition for the numerical model and then we will be trying to solve. So, this analytical method we have all discussed I am not going to the details physical method also we have discussed. So, computational method as I mentioned the solution is obtained with the help of some approximate method using a computer commonly numerical methods are used. So, wider class of mathematical formations and a do and do of fast computers. So, this was possible by the development in computer technology in last 1960s onwards. So, that is where the development of various numerical methods started and now the last 50 years many of the problems are solved using the numerical methods. So, if you go to the literature you can see that even though the basic aspects of this computer computational methods or numerical methods were developed in the end of 19th century or the beginning of 20th century. But since when we write we will try to solve the real problem it was a large number of system equations to be solved and then it was impossible by the human mind or even way of calculation or even the calculators. So, that is why the computer was always required and by 1950s after second world war computer came into picture and then first method was which was came into picture was the finite defense method then finite element method then finite volume method and boundary element method. So, number of methodologies came into picture. So, we will be discussing briefly in this course two methodologies and then related softwares depending upon the availability of time. So, in the finite defense method I will be discussing detail in the next that after this five minutes I will be discussing finite defense method. So, I am not going to the details of finite defense method in this session. So, here what we are trying to do we are having the given equations we are the variations are represented with respect to a set of points or grid of intersecting lines. So, this details we will be just after few slides it will be coming the finite defense method we will be discussing detail. And in finite element method what we are trying to do the region of interest is divided in a much flexible way and not set which values of function is found to lie on a grid system or a mesh system and then we are putting the boundary conditions to solve the system this also we will be discussing in detail in the afternoon. And then some of the latest techniques like boundary element method it is not so late but maybe 20 years or 25 years all. So, here the partial differential equations are represented as in terms of an integral equation related to the boundary and then we apply technique called Green's integral theorem which was developed by the end of 19th century and then we are trying to represent what is happening within the domain with respect to the boundary variation. So, this is the technique of boundary element method. So, this is also very much used nowadays. So, we will be discussing in detail two methodologies finite defense and finite element method and its related software for water resource planning and management. So, in all this when we are using the numerical methods there is you can see that there are three steps one is first one is called a preprocessing preprocessing means say we have to say set the domain and then we have to discretize if it is finite defense of finite element method we have to discretize the domain and we have to set the boundary conditions and we have to set the initial conditions. So, that is the step so called a preprocessing and then second step is processing the processing what we are doing. So, we are writing a computer code we are writing with respect to that computer code you will be say running the when we run the that computer code say that the original governing equations we are trying to solve with respect to given conditions. So, that is so called a processing and then the last step once you process once you run the model what is the computer gives a number of numbers only. So, that numbers if you give directly the clients your client will not understand anything. So, you have to process it such a way that you can create graphs you can create conduits you can create animations you can put it appropriately in tabular form. So, this third step is the post processing. So, in any kind of model we have to preprocess then processing then post processing. So, with respect to this say now before going to finite defense method if you want to have any few minutes discussion we can have it if you have got any questions we can you can ask.