Thanks Joseph L. Shomberg. I have a limited view of the Lebesgue integration but it seems to me it is  a much more useful(faster?) technique when confronted withtwo end point conditions for a differential equation especially when there is no clear analytical solution. Basically it would appear(subject to any correction) that we can build up solutions taking horizontal slices instead of vertical slices.If it is not a correct view it would be nice to know the correct one. The problem arose in trying to solve Navier stokes electromagnetic coupled equations of a a plasma engine where the inlet conditions and exit conditions are specified. I am looking for a numerical algorithm

Thank you once again for your interest.

 Cheers

Thank you Peter T Breuer  for your observations.It would appear however that there are some distinct advantages with Lebesgue integral (from a layman perusal on this subject in Google search). Interestingly a quite friendly gentle explanation is given at

quote"A Dynamical Approach to Accelerating Numerical ...

www.maths.qmul.ac.uk/~omj/integrationfinal.pdf

by O Jenkinsona - ‎2007 - ‎Cited by 4 - ‎Related articles

Numerical integration involves approximating an integral over a continuous space ... If f : [0,1] → R is continuous, then its integral µ(f) with respect to Lebesgue .."unquote

,this might answer your question like "This is like asking "which is better? Zermelo Fraenkel set theory or Von Neumann set theory"?" (as this is beyond my ken)

Cheers

Thank you Peter T Breuer  , for  your nice clarification  but  what I meant by "better" is precisely that it might be computationally advantageous which seems to be so in the example that is found in the reference..above. thank you once again for your patient and valuable reply...

 Cheers

Peter T Breuer   Thank you once again .. it was indeed very illuminating..especially

"It is not computationally advantageous. You can't "do" Lebesgue integration computationally."... perhaps this is the reason there are no special Lebesgue packages in scilab or matlab etc. nice..rests my quest...

Cheers

The differential equation can be well-behaved, but have singular solutions-but there isn't any way to state that one is performing Riemann vs. Lebesque integration of the equation. The sentence doesn't make sense. If the solution has singularities and one would like to compute quantities that involve *its* integral, or integrals of functions of the solution, *then* the issue of what is the useful measure on the space of solutions to the differential equation can become meaningful-i.e. what initial or boundary conditions lead to solutions that are Riemann integrable or to solutions that are Lebesgue integrable. 

For a differential equation what matters is that the derivatives of the solution should exist, therefore can be usefully approximated-else the solution doesn't make sense, not the integral. That's an independent question.

One shouldn't confuse here domain and range of the solution of the differential equation. If the measure on the domain is Riemann integrable, the only question is, whether certain functions of the solution are Riemann integrable, as one varies the initial and boundary conditions. 

If the measure on the domain isn't Riemann integrable, e.g. it's a fractal, then, while the approximations of the fractal will be, by construction-since such approximations do involve a finite number of points-and the numerical solution, once more, also, will be, since it's the output of a computer program, that represents data to finite precision-the putative limit of the numerical procedure, as the approximation to the domain tends to the fractal, may-or may not-give rise to a function, or functions thereof  that is integrable; and such a function can only be integrable in the sense of Lebesgue, since the limit measure of integration only makes sense in the sense of Lebesgue, in this case.

There's  a vast literature on solving the diffusion equation or the wave equation on fractal domains and how do the diffusion properties, or the scattered intensity depend on the fractal properties of the object on which the random walk or the scattering occurs. So how do the singularities of the domain  determine the singularities of the scattered intensity or the properties of diffusion (diffusion coefficient, critical exponent) and vice versa.

If you want to solve differential equations on a computer, you should clean your desk of all books on Lebegue integration and similar. In most cases the standard prepackaged algorithms are much better than anything you have time and talent to design, code, and debug yourself. It may be useful to consult a book like Numerical Recipies to learn how the algorithms work, and their limitations.

Numerical mathematics is mostly an entirely different field of mathematics than Real Analysis and similar areas dealing with infinite processes. Even an explicit finite algorithm like the Cramers rule for solving linear equations would be a laughing stock if suggested to a numerical analyst (rightfully so). 

There have already been a number of answers and discussions on this question, but I'm answering because I think everyone has missed the point.

The original question was posed presumably because Mr Ramesh would like to integrate a function numerically.  If he can show that the function is continuous, or only discontinuous at a finite number of points, then it is Riemann integrable.  If he can't integrate it analytically, then at least he can break it down into integrals over each range for which is is continuous, and apply any of the standard numerical integration schemes (Simpson's rule, Gauss quadrature, etc).

If he can't demonstrate that it is continuous, or if it is clearly infinitely discontinuous then there remains a real question as to whether it is integrable.  What does that mean?  If it is truly an anarchic function, then finding the area under the function might be a meaningless thing to do. 

Lebesgue integration is a technique in real analysis to determine whether a function is integrable - it extends the power of Riemann integration to include functions that are infinitely discontinuous, but for which area under the curve is still meaningful - it was the mathematical basis for establishing that the Dirac Delta was a valid tool in physics.  I'm not sure that it is a helpful concept in this particular case.

If Mr Ramesh has good grounds for believing that his function f(x) is integrable, despite being infinitely discontinuous, then perhaps an appropriate way forward would be to use Monte Carlo method, selecting random points x,y and testing to see whether y> or < than f(x).  There should come a point where adding more random points would very little difference to the integral value. If it does not converge with a reasonable number of points, then the function would not be integrable (at least in a reasonable numerical way with today's computer resources).

These lecture notes, http://www.math.ubc.ca/~barlow/preprints/stfr.pdf might be useful. 

And this paper: https://hal.archives-ouvertes.fr/hal-00003628 might be useful for setting up numerical examples-it uses finite elements. 

An example using (pseudo)spectral methods may be found here: http://lsec.cc.ac.cn/~cjxu/papers/SNA09.pdf

and here: http://homepages.uconn.edu/fractals/wave/Chan-Ngai-Teplyaev_revised.pdf

Alison> Ramesh would like to integrate a function numerically

In the question the integration of ODE's is specifically mentioned. But there is a large class of ODE problems where I think your suggestions are very appropriate: The integration of stochastic differential equations (with f.i. white noise coefficients). Of course, in such cases the Monte Carlo method would apply equally much to a specific realization of the coefficients of the equation as to evaluation of a specific solution. But in such cases the derivative of any solution is unlikely to be Riemann integrable, but likely to be Lebegue integrable.

The Monte Carlo method, applied to differential equations requires further qualification to make sense, either theoretically, or numerically, since one must specify just what space one is sampling, what one expects the Monte Carlo time averages to converge to and so on. So it is often the case that it's useful for such problems to translate the problem of solving the differential equation to a variational problem.

I would like to thank individually  all the authors above for their kind answers and views.Indeed it is overwhelming to be introduced to so many illuminating and fascinating concepts..and a pleasure..For what it is worth and with  due apologies for any errors in thinking my question was based on a purely engineering notion .which I may state as follows:

1. Suppose we have the classic" ball throw" problem described by a simple ODE and we want it to hit a particular target. we can try (numerically) various initial conditions to achieve this by trial and error using say Runge–Kutta or Simpson rule ..(handy tools for engineers) .

2. An alternative based on a wiki explanation would be to fix the end points and assume horizontal bars (ie dy instead of dx) to fit the ODE and obtain a limit which seems more attractive .

I would be glad to receive gentle  views on this .

3.The following is a nice quote from http://en.wikipedia.org/wiki/Lebesgue_integration

quote"Lebesgue's approach to integration was summarized in a letter to Paul Montel. He writes:

I have to pay a certain sum, which I have collected in my pocket. I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. This is the Riemann integral. But I can proceed differently. After I have taken all the money out of my pocket I order the bills and coins according to identical values and then I pay the several heaps one after the other to the creditor. This is my integral.

—Source: (Siegmund-Schultze 2008)

The insight is that one should be able to rearrange the values of a function freely while preserving the value of the integral. This process of rearrangement can convert a very pathological function into one which is "nice" from the point of view of integration, and thus allows for such pathological functions to be integrated."unquote

4 The intuition given in same from same source..(graphs at site)

quote"Intuitive interpretation

  Riemann-Darboux's integration (in blue) and Lebesgue integration (in red).

To get some intuition about the different approaches to integration, let us imagine that it is desired to find a mountain's volume (above sea level).

The Riemann-Darboux approach

Divide the base of the mountain into a grid of 1 meter squares. Measure the altitude of the mountain at the center of each square. The volume on a single grid square is approximately 1 m2 × (that square's altitude), so the total volume is 1 m2 times the sum of the altitudes.

The Lebesgue approach

Draw a contour map of the mountain, where adjacent contours are 1 meter of altitude apart. The volume of earth contained in a single contour is approximately 1 m × (that contour's area), so the total volume is the sum of these areas times 1 m.

Folland[1] summarizes the difference between the Riemann and Lebesgue approaches thus: "to compute the Riemann integral of f, one partitions the domain [a, b] into subintervals", while in the Lebesgue integral, "one is in effect partitioning the range of f ".unquote

Cheers

While the statements about the idea behind the definitions of the Lebesgue and Riemann integrals might be useful to get an intuitive idea, solving a differential equation involves doing something else, obtaining from knowledge of the values the derivative(s) of a function can have, knowledge about the values of a function. So it's a different problem.

Thank you Stam Nicolis ..The Monte Carlo method suggested by Alison J Mcmillan(thanks) quote " perhaps an appropriate way forward would be to use Monte Carlo method, selecting random points x,y and testing to see whether y> or < than f(x). There should come a point where adding more random points would very little difference to the integral value." uquote  seems attractive.and possibly yield results (hopefully good). Actually in the application in Navier Stokes equation we do encounter singularity in the range of mach=1 when a matrix becames singular so the equations were valid above and below this value but by an artifice (not based on solid proof  )  a solution through the singularity was obtained. but this is out of scope of the question ..

 Cheers

Narasim> I would be glad to receive gentle views on this.

The first method is known as the "shooting method". It works quite well when you want to hit a one-dimension target (i.e., specified by one real number) by adjusting just one number, like the initial velocity. Since it becomes equivalent to finding the zero of a continuous function. The advantage is that you need very little resources in terms of memory and computer code. And the computing time is reasonable, even if you have to make many attempts before hitting the target to acceptable accuracy. And in some cases it becomes impossible to hit the target at all. You always end up too high or too low, because you can only adjust the initial velocity to the numerical accuracy of your floating point numbers (that can be amended by using multi-precision floating point numbers).

However, if you try to hit a multidimensional target by choosing the required number of initial conditions, you may very soon run into an impossible task. It is worse than hitting N targets with N guns simultaneously (because aiming with one gun will affect the behavior of all the others).

Then a numerical method where you determines the full path in one go is the only available option. This requires the solution of a large number of equations at the same time (maybe even nonlinear). The finite element method is one example of such approaches, but it is usually more sophisticated that approximating the path by a staircase (a set of horizontal bars), as you indicate. The true path is likely to be very smooth, and you build this assumption into your solution method. Hence, the way to go is not from Riemann to Lebesgue integration, but to a more sofisticated version of Riemann integration.

In the case of stochastic differential equations, where Lebesgue integration may be a formal necessity (as I mentioned in a previous post), that part is actually done analytically (over short time intervals).

If you have a real problem to solve, you should check Numerical Recipes or similar books.  And try to use preprogrammed library routines, which have been developed and debugged since the invention of digital computers.

Thank you Kåre Olaussen .. you echo the method normally an engineer would use..and yes using standard methods has advantageous of a solid well built foundation on convergence and perhaps also accuracy . Regarding multi dimension application I am reminded of a  method suggested in a MIR publication that the fastest way to set amachine with many parameters is not methodically but as suggested by Alison J Mcmillan by means of toss of a coin to increase or decrease a parameter. If my memory serves me right I think it stated any number of parameters above 11 would yield a fast solution .. .In the one dimensional case I was thinking of evaluating from both endsusing simple methods as you stated and identifying the horizontal strips at both ends.and then integrating to get the area under a curve of the trajectory that becomes constant. assumption is the trajectory curve is convex.

Thank you once again for your continued interest.

Cheers

@Kåre Olaussen I would like to add the following reasoning that compels me to believe in Lebesgue as a possibly more attractive numerical solution. In Lebesgue it would appear that at every step we are using the data at the two ends , not waiting for complete integration to find the error in the target .. seems intuitively " better". I am alsotaken up by Peter T Breuer 's idea..that Lebesgue means just turning the axis through 90 degrees.and it becomes Riemann . would there be a real transformation thatdoes that using perhaps the ideas of eigen rotation in linear algebra.?

Cheers

@Peter T Breuer Thank you  for a very elucidating explanation and which I do value very much and wish that I had the proper training in pure maths  to respond well . I know the power and beauty of maths although I am an engineer. and quite often i do the unthinkable.. maybe there is a quantum/biological based computer in the future which might be able to tackle the problems where space/time mixing problems can be tackled effectively and could possibly be an answer to the issues raised but then I am straying and wool gathering..the interaction is indeed intellectually pleasant.. Thank you once again.

 Cheers