The Numerical Solution Of Differential Equations

The Numerical Solution Of Differential Equations

Statistics is an important branch in academics and it is a lot more than graphs and curves. This field is supplemented by a set of OEDs or Ordinary Differential Equations for the layman’s head.

Over the years, many mathematicians and statisticians have worked out methods to provide an adequate numerical solution of differential equations.

Here we talk about a few of such methods.

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Leonhard Euler

Euler Method

The Euler method was one of the earliest theories in practical maths. It was put forward by the stalwart, Leonhard Euler in his classic treatise Institutionum calculi integralis published in the 18th century. According to this method, the local error is proportional to the square of the step size.

On the other hand, there is the global error which is proportional to the step size. This can be derived through a geometrical description method.

A pendulum

Geometric Integration

There is a whole set of Geometric Integration methods to solve differential equations.

It makes use of geometric properties measuring the exact flow of an OED.

It is usually explained using the flow of a pendulum (a favorite example amongst theoretical physicists). The moving frame method and Hamilton’s equations are further methods that can be used for derivation.

Backward Euler Method

Euler has been such a genius and his theories spawned so many multiple related theories in the past. Here we have another method named after him. The Backward Euler Method is also popularly known as implicit Euler method. It is a quite basic numerical solution to differential equations.

According to mathematical terms, the method yields order one in time. It is called Backward Euler method as it is closely related to the Euler method but is still implicit in the application.

This can be further explained if you look at the derivation of this numerical solution from the classic standard Euler method.

Exponential Integration

Exponential Integration

These particular sets of integrators are helpful in solving differential equations (initial value problems if we talk in particular terms). Certaine and Pope were the experts who made its use widespread from the 1960s onwards. In that sense, it is a quite recent method compared to the other examples in this category.

Both explicit and implicitly constructed exponential integrators can help in solving ordinary differential equations. In addition, they can also be used for solution of partial differential equations (including both hyperbolic and parabolic examples).

Quantized State System Method

Quantized State System Method

These are often abbreviated as QSS methods and are as the name suggest, based on the concept of state quantization. The DEVS formalism has played a major role in their formation and capacity to solve differential equations.

Their greatest advantage is that they can simulate sparse systems with frequent discontinuities. Frequent research in QSS methods has made experts categorize them in three main categories: QSS1, QSS2 & 3 and Backward Quantized State System Method.

These QSS methods have been useful for software equations too later on. The QSS methods can work as numerical solvers in such cases.

Follow these and you may find your solutions!

Alien Existence Vs. NASA (Classified Information)

Alien Existence Vs. NASA (Classified Information)

Alien Existence Vs. NASA (Classified Information)

Do Aliens Exist? It has been an ongoing controversy and the reason for many debates. However, two events have revealed some new aspects in the Alien Existence Vs. NASA because of what “Anonymous” said. Apparently, the hackers broke into NASA’s database and found some classified information. After this, social media was flooded with news about NASA’s announcement regarding their discovery of the existence of extraterrestrial life. It was supposed to be classified information that was out in the public arena.

This is what many people believe aliens look like, but they may not look so exciting in reality.

Anonymous Hacktivist Group 

However, it is a controversial piece of news because it was revealed later that the announcement was not made by NASA but by the collective Anonymous. The YouTube page of the Anonymous also had videos saying things like “According to NASA, other planets have hosted aliens or other forms of intelligent lives long before Earth.”

According to the Hacktivists, NASA is very close to announcing that they have discovered alien life. They have based these claims on evidence like statement given by Thomas Zurbuchen who is the associate administrator in the board of Scientific Mission of NASA said that they are about to make the biggest discoveries in the history of humanity. He also mentioned about some of the recent breakthroughs in astrobiology that includes finding Hydrogen on Saturn’s Enceladus moon, and geyser’s on Europa’s Jupiter satellite, etc. These are some of the advances that could lead to discovering extraterrestrial intelligence or aliens.

However, NASA made the announcements that said that clarified that there are no reports or evidence about extraterrestrial life that are pending from their end. They stated that we are alone in the universe and that there are no clear indications or advances to prove the contrary.

However, the Anonymous group is very assertive about the NASA’s knowledge of the extraterrestrial life. It is possible that their claims are exaggerated, but the evidence and the excerpts of statements and speeches that they highlight from NASA’s officials could be worth pondering. The discovery of new planets and the possibility of water in some of them is also great evidence in this regard.

It is also a possibility that the aliens are so advanced that they don’t even need water. Who knows? There could be endless possibilities or theories that prove the existence of aliens and the contrary. But the statements published by the Anonymous are also interesting. Even though it may seem a little exaggerated, it would be worth thinking about it.

do you

Could there be some advancements and NASA doesn’t want to disclose it yet? Is there a possibility that they are waiting for some more evidence before they can make an official statement? Whatever be the case, I will keep you up-to-date with all the latest news on alien theory. Maybe they exist, or maybe they don’t? But until we are sure about it, we can keep speculating and keep looking for answers.

10 Things Scientists Can’t Explain About UFOs

10 Things Scientists Can’t Explain About UFOs

UFOs have always been a great mystery since the ancient times. From Prehistoric cave paintings to ancient texts mentioning about some “fire in the sky,” various sources always talked about unusually shaped and strangely flying contraptions in the air. They appear out of nowhere and whiz past one’s line of sight in a blink of an eye. The descriptions of their shapes were always the same, yet unconventional throughout the ages.

As always it has been the case, there will always be people, especially from the scientific community who will try to debunk any UFO claims. However, there have been times when even skeptics from the scientific community could not falsify claims of UFO sightings by offering alternate explanations.


Super Fast And Erratic Motion

Fighter pilots, both and during and after World War II have seen UFOs whiz past their airplanes, but not before making strange and unconventional maneuvers in the air that defy basic aerospace physics and logic. The UFOs would suddenly change the direction of their motion or re-orient themselves and fly in a different direction within less than a second.

Changing Colors

People have recorded UFOs on their cinematography cameras in the mid-twentieth century, where they changed their hues from red to blue, from magenta to green and the justification of that being gas and plasma fails to explain itself.

Allied As Well Axis Pilot Having Same Experiences

Accounts by World War II veteran pilots from both the sides have described what they thought to be advanced enemy technology, but later finding out that their enemies experienced the same.

Self-Maneuvering ‘Space Debris’

Astronauts working on the International Space Station have observed what they thought as remnants of old satellites, moving in directed and intelligent way. They changed directions and speeds after an interval.

The UFO Crash A Roswell, New Mexico

In 1947, the remote town of Roswell in New Mexico was met by a crashing UFO in its outskirts. Some say Area 51 was built to conceal the alien craft as well the dead bodies of its alien occupants.

The Hessle UFO

Women in the town of Hessle saw a UFO with three corners, in the shape of a triangle and with characteristic erratic maneuvers. Nobody has been explaining till date what possibly could that be.

Green Fireball Sightings

These happened between December 1948 and April 1955 in New Mexico and were seen by almost everybody. Some say that it was an alien reconnaissance-rescue craft for the one crashed in Roswell in 1947.

The Crop Circles

Intricate geometric patterns started appearing on fields across the USA, Australia, Britain, Japan and New Zealand since the late 1970s to early 1990s.

Alien Abductions

Apart from the real nutcases, many who claim to have abducted by aliens have passed psychometric evaluation tests for schizophrenia, psychosis and sleep paralysis. Scientists could not explain it when they shared similar abduction experiences.

Disappearance Into Thin Air

People have also filmed UFOs disappearing into thin air all of the sudden. Scientists have been divided between skeptics and believers over these claims with video-graphic evidence.

Although it is highly debated, there is good evidence to believe that there good be a relation between UFO sightings and the existence of extraterrestrial intelligence.

Are There Extraterrestrial Life Forms?

Are There Extraterrestrial Life Forms?

Are There Extraterrestrial Life Forms?

“Are there extraterrestrial life forms?” is a question that not only the scientific community but also many extraterrestrial enthusiasts have pondered upon since a very long time. Many researchers from SETI and NASA believe that life could have existed in billions of planets. There are many controversial debates and theories, but there has been no concrete evidence of the presence of extraterrestrial life forms. However, interestingly, the scientific community has not been able to discard it either completely. Therefore, it is an interesting question that our generation and our forefathers have been trying to answer.

However, with great scientific advancements happening every single day, the chances of getting more concrete evidence is higher than ever now. There are two sides to this question. You can read more about the new advancements to keep yourself informed.

In this blog, I will discuss both the sides as well as the neutral standpoint and leave you to take a stand or begin your research from a new standpoint.

The Positive Standpoint

There is one section of people who are very positive about the extraterrestrial life forms in the universe and strongly believe that they exist. Within this community, there are again some divisions among people who believe that they are more superior and advanced to us while others believe that they can be any form of bacteria. No matter what, they believe that some form of life apart from the one that we have on earth must exist in the wide universe.

The Negative Standpoint

On the other hand, there is a section of people completely discard the existence of extraterrestrial intelligence. According to them, these are all conspiracy theories, and the radiations or the unusual electromagnetic waves are a result of a certain phenomenon or changes in the celestial bodies or planetary movements. They believe that if aliens existed, they would have done something to show their existence. It is not normal to be quiet for so long. They completely discard all the theories and say that it is none of our business to start a quest for something that we are not even sure would exist. They think that it is wastage of time and energy to research on extraterrestrial intelligence.

The Neutral Standpoint

The neutral standpoint believes in waiting for the final verdict. It is possible that the aliens could be millions of light years away and therefore we have not received their messages yet. With Arecibo message, scientists tried to talk to them but the expected time to receive their message is not up yet. So we cannot completely give up hopes. Hence they don’t align themselves either with the positive standpoint or the negative one.

Final Verdict

There can be no final verdict as of now because it is an ongoing debate. Many questions have not been answered in the extraterrestrial realms like the spotting of UFOs or the “WOW!” signal or many other signals that are recorded by NASA since decades. It is possible that some of the signals could be a result of some other astronomical phenomenon, but it is not certain yet. Hence, it would be wrong to discard the theory that extraterrestrial life forms exist completely. At present, it would only be logical to keep all the avenues open for debates and discussions without necessarily concluding. We are living in an interesting time with discoveries every single day. So, maybe we are not very far from finding out the truth about aliens.

Websites Dedicated To Extraterrestrial Info

Websites Dedicated To Extraterrestrial Info

Websites Dedicated To Extraterrestrial Info

Extraterrestrial researchers believe that there is another form of life in the universe which is different from the humans on the earth. There are various theories about the extraterrestrial forms and life. Some believe that they are more advanced than humans while others believe that they could be just any other form of bacteria. The presence of extraterrestrial lives in the universe is a matter of immense debates, and there are constant researches and breakthroughs in the scientific community. It is sometimes hard to keep up with the latest events since a lot is happening in the field all the time. With the advancement in technology, scientists are getting better at deciphering certain phenomenon. Hence, if you want to be updated with the leap forward in this field, you can follow some of the websites that could serve very informative for you.

Websites dedicated to extraterrestrial information are great for the enthusiasts like you. These websites provide with complete information about the latest research and breakthroughs in the field. Some of those websites are:


Search for Extraterrestrial Intelligence (SETI) is a website from the SETI Institute that is dedicated to finding information about extraterrestrial breakthroughs. The mission of the institute is to dig deeper into the understanding of the nature and origin of life in the universe. They aim to use this knowledge to guide and understand if there could be other life forms like ours in the universe. The website is regularly updated with the advancements and innovations in the field. The main aim is to share knowledge and collaborate with the scientific community, people, media and the government.  You can regularly visit this website to keep yourself updated with all the latest updates in this field.


Another interesting website to keep you updated with the latest happenings in the extraterrestrial world is It is a one-stop website for all the happenings in space and also holds a section called “search for life” where it exclusively discusses all the happenings and new ideas in the search for extraterrestrial life. You can find some stunning photos of the galaxy as well as all the news about rocket launches and new space experiments. You just need to visit this website once in a while, and you will get all the news and updates within minutes. You can also subscribe to its newsletter to get regular updates on your email, so you don’t miss out on any important innovation. It is a great resource for all you enthusiasts.

MUFON.COM has the mission to carry out an in-depth investigation of all UFO sightings and make sure no unusual phenomenon go unnoticed. Moreover, it aims at promoting research and funding on UFOs and keeps the public well informed of all the latest happenings. You can join as a member by paying a little amount, and in return, you get access to their e-Journals and all the latest updates. You have an option to choose from various types of membership. You can even volunteer with them if you want to enhance your understanding or get a first-hand experience in the field. It is a platform that believes in promoting more research and including more and more people in the quest to find other life forms in the universe. You can visit this website, and you will not be disappointed.


NASA is a website that you need to visit as an enthusiast of extraterrestrial life forms. It is no surprise that you will get live updates on their findings and research. You can even find some interesting thought-provoking photos on their website. Being a regular visitor of this website will help you keep up-to-date with all the latest happenings in the realm of the space.

You can bookmark the websites mentioned above to keep yourself updated. Of course, if you are a regular visitor of my blog, you will get all the latest news along with some thought-provoking discussions with a community of highly enthusiastic people. With this blog, I aim to bring all the available information to one platform in a lucid manner. A highly motivated community in this blog also makes it special because it is always open to debates and discussions. I hope to make your quest for extraterrestrial research more interesting.

Extraterrestrial Intelligence And Modern Scientific Discoveries

Extraterrestrial Intelligence And Modern Scientific Discoveries

Extraterrestrial Intelligence And Modern Scientific Discoveries

The Arecibo signal is one of the most remarkable scientific leaps taken by humanity in its attempt to establish contact with aliens. The radio signals pack up basic information about humanity inside it such as prime elements found on the planet, our Solar System and Earth’s location inside it and the basic structure of humanity and its genetic composition.

However, this was just a small (but significant contribution) in the vast array of radio signals that humanity has been emitting into the open outer-space since the advent of mass communications. With radio waves from television signals, satellite broadcasts and other sources crossing Earth’s atmosphere in the vastness of deep space, humanity making daily attempts to communicate with a sentient species out there.

Let us take a brief look at what arrangements we have made to establish contact with intergalactic extraterrestrial intelligent.

The SETI Institute California


Search for Extraterrestrial Intelligence has been operating since the 1970s and has deployed the Alien Telescope Array that is dedicated to catching radio frequencies emanating from various sources in the outer-space and weeding out the ones that were likely sent by sentient species.



Arecibo Observatory

The Arecibo Observatory is CURRENTLY home to the world’s largest single-aperture radio telescope with a diameter of 1000ft (305m).

Arecibo Observatory

Located in Puerto Rico, this was the radio transmitter that sends the Arecibo signal mentioned above.


Transiting Exoplanet Survey Satellite helps pin down Earth-like planets in remote solar systems by measuring the amount of light that they obstruct from their star when they complete one rotation.

 These and many others have helped us to gain a lot of insight into the dynamics of cosmic phenomena that can hint towards a possible existence of life or even sentient life.

The existence of the planet Gliese 581g which is twenty light years away from the Earth has been a groundbreaking discovery in our quest for finding extraterrestrial life.

However, the closest we have come to finding a possibly habitable planet is the discovery of Proxima B, which is just 4.2 light-years from the Earth in the Proxima Centauri star system, has a mass 1.2 times of Earth and is similar to our planet in other astronomical parameters.

However, the closest we have come to possibly finding water on other planetary body is inside our Solar System on Europa one of Jupiter’s moon when Hubble Space Telescope discovered plumes of water on the satellite. Let us hope that there is an ocean of liquid water below with bustling life below inside it.

These discoveries especially the former two have been crucial in the further discovery of alien life forms on other planets. However, to be honest, finding life-supporting planets has one in a millionth chance, finding sentient life falls to several magnitudes below.

Let us hope that if an alien civilization is advanced to a level of three or beyond on the Kardashev scale, they will honor our attempts to establish contacts in the vast reaches of deep space by responding cordially.