Mathematics - Science topic

Three more papers published by my purported to prove Fermat's Last Theorem are attached herewith.

Rashmi Yadav ji,

What do we understand from the expression 'Vedic mathematics'?

Many Sanatana and Arya Smaj followers recite Veda even today. Are we still in Vedic period?

Vedic period, post Vedic period are expressions coined by ignorant foreigners for their convenience to understand History of people living in this region. They called them (us) Hindu.

I prefer dividing the period in two segments

1. When Veda were were written and updated from time to time.

As of now we are not sure when our sages started compiling the three Veda; Rug, Sama, and Yajur. A rough estimate is around 50,000 years BP. The last update in these three Veda was made by Maharushi Veda Vyasa, around 4500 BCE. He also compiled one more which we know as Athrva Veda.

I have sound reasons to believe that during this period ancient astronomers used very large numbers, simple arithmetical operations like addition, subtraction, multiplication, and division.

2. Period when there no more addition to four Veda, Post 4500 BCE.

We have internal evidence in Indian scriptures that around 2500 BCE, astronomers fixed the ratio of the square of circumference to the square of the diameter of a circle as equal to 10. They also started using sine and inverse sine tables during this period. These tables were of a limited number of angles which came up frequently in day-today astronomical calculations.

Much of the advancement in mathematics was done by Kerala school of mathematics in the CE era. A number of mathematical expansion series were developed by these mathematicians/astronomers.

Per my understanding, Vedic mathematics is like our ordinary arithmetic. Ancient astronomers had developed sutra/ready reckoners to complete complex arithmetical calculations with ease. In Ancient Indian astronomy one comes across 'rule of three'. This we know as the problems of ratio and proportion. When three quantities are known, the fourth is calculated by using this method. The oldest reference, I have seen this formula is in Vedanga Jyotisha. Many scholars have dated this book on ancient astronomy to around 2000 BCE. My findings suggest that this is about 35000 years old.

I shall be glad to answer any supplementary queries.

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Everything has two sides dis/advantages. Please look at Table 3 and Table 4 of the attached file.

Clarivate announced that starting with 2023 ESCI-indexed journals will also be assigned an impact factor. See: https://clarivate.com/blog/clarivate-announces-changes-to-the-2023-journal-citation-reports-release/

23 November MMXXII

No.

Cordially...

ASJ

There is even such a discipline called "COmputational Social Sciences"...

I recommend research results from authors of the French group of Didactis of Mathematics as: Vergnaud, Bideaud, Meljac, Fischer, Brun. Also, from Oxford University Peter Bryant and Terezinha Nunes.

Thank you, I have got some idea.

Anybody have a matlab code for this article

"An improved opposition-based marine predators algorithm for global optimization and multilevel thresholding image segmentation"

Most statisticians agree that the minimum sample size to get any kind of meaningful result is 100.

We can imagine any finite quantity to be halved, and then each half to be halved again, and so on, without end. You can then say the finite quantity has infinitely many parts. It's just a conceptual distinction.

Both software solutions are superb. Next, many great suggestions regarding graphics software solutions you will find in this post:

Raphael Neelamkavil , you said, "But in your paper "The zero-dimensional physical theory (V): information, energy, efficiency, and intelligence" you exhibit enough awareness of the above facts. Hence, I do not understand what else you meant by the question at the end of your comment!".

The title of this forum topic is, "Can the Primitive Notions (Categories) and Axioms of Mathematics, Physics and Philosophy Converge?". When I said, "This is all a good question and debate, yet what is deliberate misinformation and ignorance, what is the drive for it, and can it be written about in a way so as not to promote it as being useful?" I am essentially asking whether or not the primitive notions are already granted by the contemporary ideas of mathematics, physics, and philosophy. My work highlights they are not, so a clear issue in this debate is all about what is assumed and what is not. My work with zero-dimensionality assumes nothing, and thence creates a new spectrum of ideas for mathematics, physics, and philosophy which has been, is, and perhaps still will be for most.

In Leon Lederman's wonderful book, "The God Particle" he comments about a pecking order in physics. It goes something like, "the experimental physicists only defer to the theoretical physicists who in turn only defer to the mathematician and the mathematician only defer to God..."

Physics is not mathematics. The role of physics like all sciences is to understand the universe. The theoretical physicists' role is to develop theories that explains all the experiments from the past plus allows predictions to be made to test the and potentially falsify the theory. The role of the experimental physics is to develop and perform those experiments. There is often as much deep mathematics used by an experimental physicists as a theoretical physicists.

The role of the mathematician is simple to expend the understanding of mathematics through rigorous logical reasoning. If a physicists or any other scientist finds these tools useful - then great.

Arguably the greatest theoretical physicists of the second half of the 20 Century was Richard Feynman. Here is what he said about physics.

"It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong."

I would suggest for a better understanding of physics - what it is and isn't - a read of "The God Particle" would be recommended

https://en.wikipedia.org/wiki/The_God_Particle_(book) .

Clarence Lewis Protin `Writing "literate" .agda files in which code is mixed with an informal mathematical presentation of the formalisation seems the best option !`

There are software development practices akin to this. In the open-source projects that I have participated in (other than my own), comments are eschewed. There are variations on Literate Programming that some adopt, although it requires more mechanical support. In software, commentary helps differentiate what the software is for in contrast to what it is, establishing how the software is an interpretation of some conceptual model.

The risk used to justify the elimination of commentary is the problem of maintaining consistency between the commentary and the code as it is developed/maintained.

I have no objection to this approach, nevertheless. I find the commentary important for my own recollection of the purpose of some code and confirmation that the code accomplishes that.

I do not see how this methodology is a solution to the problem raised in this question though, unless it would be useful in maintaining the proof in the face of breaking changes up-level or down-level. Please say more about that.

As far as I understand your problem, you first need a mathematical relation between the instructions and rank. For instance, Rank x should correspond to some instruction value as y and vice versa; it means you require a mathematical function.

So there are various methods/tools to find a suitable (as accurate as you want) to find mathematical function based on given discrete values like curve fitting methods or the use of ML.

Further, Once you obtain the mathematical function, run your code a few times, and you will get a set for various combinations of (instruction, rank). These set values will work as the feedback for your derived function. Make changes based on the feedback, and you will get a much more accurate function.

I hope you are looking for the same.

One possible approach is presented in the attached word document

And this:

"Theoretically, such a system can be solved exactly; i.e. it can be determined whether the system has real solutions or not, and if it has, its solutions can be calculated exactly." is exactly what CAS-systems like Maple and Mathematica do. Good for Dudley J Benton to have a perpetual subscription. My experience is: the Maple syntax is simpler - and the sysmbolic solution of partial differential equations is better in Maple than in Mathematica. But for people wthout a license Mathematica is easier to rech - via wolframaplha.com. And there you need not to adhere to the syntax rules.

For vectors X1=[x1,1;x2,1;..,xn,1]; X2=[x1,2;x2,2;..,xn,2];,.., Xn=[x1,n;x2,n;..,xn,n]

M=[m1,m2,..mn] - vector median,

where

m1=med(x1,1; x2,1;..,xn,1)

m2=med(x1,2; x2,2;..,xn,2)

- - - - - - - - - - - - -

mn=med(x1,n; x2,n;..,xn,n)

I did give a physical explanation for why your boundary condition enforces the function f to become zero at r=0. In at least one physical interpretation, it models a system with infinite emission/absorption rate at r=0. This forces the temperature at this point to immediately become equal to the environmental temperature. In this interpretation f is (proportional to) the difference between body and environment temperatures. When you define a physcally unreasonable it is only fair that it provides you with physically unreasonable results ;-D

Thank you, Anton Vrdoljak.

For my opinion we can use any product formulas only under condition of independence. For example the formula that you used P(a,b│c)=P(a│c)∙P(b│c) is valid if a and b are independent each from other. Both may be dependent from c. But it can be the case of so called "false dependence". For example a is the usage of energy, b is sale of warm clothes, c is decreasing of tempetature.

Sorry, Jianhing. But I think you have misunderstood something in the lecture. Frequentist statistics, which is an interpretation of probability to be assigned on the basis of many random experiments.

In this setting, on designs functions of the data (also called statistics) which estimate certain quantities from data. For example, the probability p of a coin to land heads is given from n independent trials with the same coin and just counting the fraction of heads. This is then an estimator for the parameter p.

Each estimator should have desirable properties, as unbiasedness, consistency, efficiency and low variance and so on. Not every estimator has these properties. But, in principle one can proof, whether a given estimator has these properties.

So, it is not a characteristics of frequentist statistics, but a property of an individual estimator based on frequentist statistics.

The solution is not a mathematical formula!

The question can be approached by two ways:

a) experimental: prepare homogenous standards having the same density and geometry as your samples, #1 material with low K,Th,U content, #2 same as #1 with known amount of K added, #3 same as #1 with known amount of Th added ,and #4 same as #1 with known amount of U added.

From the differences between measurements #N-#1 you may deduce the contribution of each of the three elements to the spectrum and, more specifically, the contribution of each of them to three regions of interest corresponding to strong peaks of each element, usually 1.46 MeV for 40K, 2.61 MeV for Tl208 (progeny of Th) and 609 keV for Bi214 (progeny of U), assuming secular equilibrium in Th and U families, or at least the same disequilibrium conditions in sample and standards.

It is then possible to calculate a matrix relating the measurements in the 3 ROIs to the three activities.

b) Monte Carlo simulation. The NaI detector is usually quite simple and can be simulated with a good accuracy. Experimental measurements are still necessary to evaluate the gaussian broadening of the peaks at the different energies, and an experimental validation of the MC model is always desirable.

If the methods are different, it is likely that there is no correlation ...

However, it would help a lot to know the datasets and the processing you say you have done.

My predisposition to mull with numbers and mathematical equations consistently, even though I do not specialize in the math field as a professional, is due to having FUN filled mentors early in life. They were so creative to attach images and anecdotes to math problems that my sense of wonder and fun was constantly tickled as I trundle along mathematical mines and problems. Missing the correct answer was not occasion for embarrassment but for more opportunities to discover funny and creative ways to approach math problems with new, innovative, and fun-filled attempts. My mentors showed me the short-cut methods, the winding methods, and the funny but sure fire methods. I do not believe I ever hated math. Math is FUN!

I thought it would be better to type my work. Here is a link to the acceptable version of the article : https://www.researchgate.net/publication/364357359_A_contradiction_in_the_formula_of_Poisson_which_destroys_Riemann's_Hypothesis

I thought it would be better to type my work. Here is a link to the acceptable version of the article : https://www.researchgate.net/publication/364357359_A_contradiction_in_the_formula_of_Poisson_which_destroys_Riemann's_Hypothesis

Therefore, you have 20x20 = 400 control points. If you do georeferencing in Qgis, you can use all control points or some of them, like every 5 Km (16-points). During resampling, all pixels have coordinates in the ground system.

If you do not do georeferencing (no resampling), then you could calculate the coordinates of unknown pixels by interpolation. Suppose a pixel size a [m], then in one km, you have p = 1000/a pixels, and therefore known coordinates have the first(x1,y1) and the last(x2,y2) pixel. The slope angle between the first and last pixel is:

s = arc-tan[(x2-x1)/(y2-y1)]. Therefore, a pixel of a distance d from the first pixel has coordinates x = x1 + d.sin(s) and y = y1 +d.cos(s). You can do either row of column interpolation or both and take the average.

Ricardo Simão Pereira Lopes , indeed, your approach is the approach of many:

My question is if you see a problem there and if so why and if not why not.

For me the following book is very useful

Functional Equations and How to Solve Them (Problem Books in Mathematics)

Springer

Christopher G. Small

Year:

2006

Language:

English

WO 2017/037729 A1: Concurrent architecture of Vedic multiplier-an accelerator scheme for high speed computing

Mathematics is essentially one of the professional tools of science, including for physicists.

Admire the tool... If the cook says "Miracle saucepan", then the food will not get better...

Students should be shown the application of mathematics from a practical point of view. On the other hand, it is necessary to respect the theoretical side of mathematics, which is not easy for everyone to learn.@Peter Kepp Rickardo Gomes

Hi fellow researcher,

I think the answer depends on what kind of equipment you have on your disposition and what is your purpose. The simpler way is to have a model blade and compare it to the blade you want to determine the integrity using the physical properties of the blade.

For example, if you have an accurate sound sensor and a device that produces an specific sound profile in the model blade you can use this device to produce a similar sound in the test blade. Comparing the Fourier decomposition of frequencies of the sound produced by the test blade will give you information of cracks and non-uniformity in density produced by the forging process or even if the material the blade was made is of quality or not.

You can do a similar approach with thermodynamics. you can heat Specific spots on the test blade. As cracked regions do not conduct heat as well as non-cracked regions you will observe anomalies in the heat map of the blade.

Notice you don't really have to calculate something, but just know what to expect with a model blade data.

The RHS is a fraction, whose numerator and denominator are quadratic expressions in Δ. Therefore the fraction takes positive values when numerator and denominator are of the same sign...

Also, Hamilton-Jacobi theory can be applied with not only fractional derivatives but also, definition of fractional derivatives and integrals on time scale Yazen Alawaideh

The question is too general. To generate the basic idea of the algorithm, you need to know the detailed statement of the problem. Typically, the process of developing an algorithm starts with the identification of your problem complexity class. If there is no evidence or clear feeling that the problem is NP-hard, then it is reasonable to try to develop a polynomial algorithm for solving it. Such algorithms are usually based on the use of specific properties of the problem. Sometimes it is possible to construct a polynomial algorithm based on the general scheme of dynamic programming, taking into account the specific properties of the problem.

If the problem is known to be NP-hard, then branch-and-bound methods, dynamic programming, and their modifications often work well for a relatively small problem dimension. Sometimes it is possible to build a successful formulation of the problem in the form of an integer programming model, followed by the use of appropriate methods or ready-made software. For high-dimensional problems, you can either use well-known metaheuristics, or develop your own approximate algorithm. In the latter case, success is usually based on the use of the problem properties. As you can see, in any case, it is useful to start by studying the specific properties of your particular problem.

Dear, Tekle Gemechu basins of attraction comes from the dynamical system and chaos theory. It is related with systems behavior. There are several example of basins of attraction. Therefore, definition and examples of basins of attraction are given in scholarpedia. Below scholarpedia's link:

Hi Hieram, youre welcome! Imho fun should be induced by fruitful scientific research supported by open commenting one other’s ideas (opposed to endless fruitless repeating discussions what presumingly not cannot ever work … until it does) like iSpace („integer-Space„ or „complex-Space“ when treating the i as the one for a complex number) able to derive and decipher inter-relationships, dependencies and calculate exact arbitrary precision numerical values for most but all constants of nature.

Also recently a new true quantum geometric iSpace-IQ unit system has been developed, able to directly represent native quantum relations of contants while keeping strictly compatible to MKSA/SI system showing a single *time* based conversion factor, effective predicting quantization of time itself.

So - no - being a true long time Apple expert consultant i’d say we do not need to fear to be sued for (at least not in the foreseeable future ;-) ). And please all take the time to read thru the very short yet imho rreally convincing math of both of my newest papers to be found on my RG home.

Here is a link to RG summary of my iSpace project:

Metric space A is said to be a totally bounded if every Cauchy sequence in A has convergent sub sequence

Theories and Theories of Truth | SpringerLink

I have many times served as an applied mathematician on a multi-disciplinary team, including, for instance, biologists, meteorologists, geologists, hydrologists, etc. The other team members possessed essential knowledge to the success of the project but not the math skills. This is good for you to recognize in the study of business because nobody knows everything. A diverse team can solve problems that a single individual cannot. I would have the biologists explain over and over again that part they understood (like fish behavior or metabolic needs) and I would create a model. Then we would run the model and ask, "Is it responding the way you expect it to?" One of these interesting projects considered production of peanut butter crackers and cheese crackers sold in vending machines manufactured on the same assembly line at a factory on the road from my house to the school. Walk over to the Mathematics Department and find a graduate student studying applied mathematics who is looking for a project. You can help each other.

Ankur Maheshwari I agree with Ben Cardoen that a heuristic/partial search algorithm may provide a sufficiently good solution to an optimization problem, especially with incomplete or imperfect information or limited computation capacity where commercial solvers may struggle. But the key issue of heuristics is that they do not guarantee that a globally optimal solution can be found on some class of problems when compared with commercial solvers such as Ipopt, Cplex, Mosek, Gurobi,..., for NLP or LP or MILP or MINLP problems.

Hi Sunday Fadugba I have no experience with SCILAB (and very little with MATLAB). What I can recommend is using Mathematica (if you can afford it; or maybe you can use the free version on Raspberry Pi) or Julia (free). Hope it helps.

I like this new thinking. But for me it is difficult to say "true" or "false" since everyone has his own standard. I feel interested in the definition convenient both in the application and mathematical analysis.

I agree that it starts in Piaget's sensory-motor stage of play. Mary Reilly's Systems Explanation of play: Learning through play and the Third form of Information Processing theories mirror and expand on Piaget's Theory of play and learning.

Dear Abdollah Aliesmaeili ,

Just based on the metrics I would suggest that you might have a look at:

Educational Studies in Mathematics https://www.springer.com/journal/10649 CiteScore 3.6 (SJR Q1 journal), Impact factor 2.853 (JCR Q2 journal)

ZDM – Mathematics Education https://www.springer.com/journal/11858 CiteScore 4.1 (SJR Q1 journal), Impact factor 2.481 (JCR Q2 journal)

Journal of mathematics teacher education https://www.springer.com/journal/10857 CiteScore 3.6 (SJR Q1 journal), Impact factor 1.786 (JCR Q2 journal)

Research in Mathematics Education https://www.tandfonline.com/journals/rrme20 CiteScore 3.4 (SJR Q1 journal), ESCI indexed (so no impact factor and Q ranking (yet))

Best regards.

Dear Shireen,

There are data-driven and parametric that you can utilize to detect chaos. If the system is well-defined, a good starting point can be Lyapunov Exponent and Poincare map. If you have only output of the system then you need data-driven methods like 1/0 test or entropy-based methods.

Regards

coupling equations available in ODEs and also built-on ode45, ode15, ect code in matlab.

By practical applications

Thanks a lot, dear Javad Fardaei, for the article

Dear Sandip,

to answer your question, I agree with Abdelghani: you have 1) to transform the 2nd order equations into 1st order ones and then 2) to use a MATLAB ode solver to numerically solve the resulting 1st order system.

To do step 1), you have to introduce new variables that correspond to first derivatives. For instance, let us say that you have a system of two 2nd order ODEs in x=x(t) and y=y(t), e.g. x''=f(t,x,y,x',y') and y''=g(t,x,y,x',y'). Then you can introduce two dummy variables X=x' and Y=y' (for example) and get a system of four 1st order ODEs: x'=X, X'=x''=f(...), y'=Y, Y'=y''=g(...).

Then, in order to apply step 2), you have to define a variable vector, say v=[x,X,y,Y], and a function, say myFun, that defines the system. In this case:

Then, choose an ODE solver (for example ode45) and solve the system by using the basic command [t,v] = ode45(@myFun,tspan,v0); where tspan is an interval for the independent variable (e.g. tspan=[0,10]) and v0 is a vector containing the initial values for x, X=x', y, and Y=y'. So please notice that you must provide initial values for the first derivatives!

You can find more details about all this stuff here: https://www.mathworks.com/help/matlab/math/choose-an-ode-solver.html

Hope this helps! :-)

Optimization is a broad field of mathematics and not some mysterious quackery. I am not surprised that someone from another field entirely (wireless communications) might not be familiar with these topics. This is why I stress the importance of multi-discipline teams. Nobody knows everything. If you get smart people from different fields together in the same room to discuss a problem, you may find some clever solutions that any one person alone would never come up with.

From my experience, I would not use a linear predictor for elasticity. I know your question is asking about mathematically correct solutions but I don't think that's your issue. I would probably use logistic regression as a first choice.

An activity theory perspective on contradictions in flipped mathematics classrooms at the university level, Helge Fredriksen and Said Hadjerrouit, 520-541, 2019

If the distribution of the zeros of the Riemann zeta function is implicit in your question, then you may find the following paper exciting:

Regards,

Hi, it should be taken into account that it is actually 'stress' doing the deformation for us. So when you use a roll with a smaller radius, the contact area between the roll and the sheet/foil is also smaller, increasing the stress.

Stress = Force / Area

Look at the attached screenshot search and I think your question is answered if I understand you. Best wishes David Booth

Thank you, Dr. Len

Please clarify your question to help you.

One can construct simple model (Shifted Harmonic Oscillator model) and derive tractable equations to obtain absorption and emission spectra.

1. Absorption and Emission Lite: Assume ground and excited states are shifted classical Harmonic Oscillator. Ground state |g> energy: E_g(R) = 1/2 ω²R² & Excited state |e> energy: E^_e(R) = 1/2 ω²(R - R₀)² + ΔE. Treating R classically one gets absorption (I_a) and emmision (I_e) spectra as:

I_a(e) = exp[(e-ΔE)²/2σ²] I_e(e) = exp[(E-ΔE + ω²R_0²)²/2σ²] (Arbitrary Units)

absorption and emission is peaked at different energies and the difference is ω²R_0² --- stokes shift, see: https://en.wikipedia.org/wiki/Stokes_shift

σ is the linewidth caused by molecules coupling to environment (Homogeneous Broadening) caused because individual molecules see slightly different environment, e.g. ΔE (Inhomogeneous Broadening) is different for different molecule.

2. Absorption and Emission Pro: Same as 1, but treat R quantum mechanically,

this gives |g>|v₀> , |g>|v₁> ... |g>|v^_n> states and |g>|v₀'> , |g>|v₁'> ... |g>|v^_n'> where |v^_n> is nth vibrational state centered around 0 and |v^_n'> nth vibrational state centered around R₀.

With this we write the absorption at T = 0K ( or ℏω >> kT)

I_a(e) = ∑_n |<v_n'|v₀>|² exp[(e-ΔE - nℏω)²/2σ²]

= ∑_n (1/n!) * (ωR_0²/2)ⁿ exp[-(ωR_0²/2)²] exp[(e-ΔE - nℏω)²/2σ²]

I_e(e) = ∑_n |<v₀'|v_n>|² exp[(e-ΔE + nℏω)²/2σ²]

= ∑_n (1/n!) * (ωR_0²/2)ⁿ exp[-(ωR_0²/2)²] exp[(e-ΔE + nℏω)²/2σ²]

|<v₀'|v_n>|² are Franck-Condon factor,

|<v₀'|v_n>|² = |<v_n'|v₀>|² = (1/n!) * (ωR_0²/2)ⁿ exp[-(ωR_0²/2)²].

3. Absorption and Emission Pro Max: Same as 2 but at finite temperature T. Assuming instant thermalization --> states are population according to Bolzmann distribution,

I_a(e) = ∑_m∑_n |<v_n'|v_m>|² exp[(e-ΔE - nℏω)²/2σ²] * P_m

₌ ∑_m∑_n |<v_n'|v_m>|² exp[(e-ΔE - nℏω)²/2σ²] * (exp[-βmℏω] /∑_kexp[-βkℏω])

= ∑_m∑_n (1/n!) * (ωR_0²/2)ⁿ exp[-(ωR_0²/2)²] exp[(e-ΔE - nℏω)²/2σ²] * (exp[-βmℏω] /∑_kexp[-βkℏω])

I_e(e) = ∑_m∑_n (1/n!) * (ωR_0²/2)ⁿ exp[-(ωR_0²/2)²] exp[(e-ΔE + nℏω)²/2σ²] * (exp[-βmℏω] /∑_kexp[-βkℏω])

P_m = (exp[-βmℏω] /∑_kexp[-βkℏω])

In perfect graphene, the D-peaks are by symmetry not Raman active. ideal graphene should only show the single G-peak. However, when there is anything disturbing the symmetry and structure, the condition is relaxed and the D-peaks become visible. Hence, their name D for "defect" peaks. Based on the relation of peak intensity or the integrated intensity below the modes, the quality of graphene layers can be quantified.

There are numerous reviews on Raman spectroscopy in graphene, explaining which information can be extracted and how they can be extracted from analyzing the spectra.

Check, for example, this article by Andrea Ferrari:

You should be able to quickly find numerous more articles.

If you can send me a sample data file, I can have a look at it...

Thanks for this

Thank you Masuduzzaman.

Hello Nasiru,

What's the reason for wanting these values? That may have more to do with what sort of approach, if any, might make sense.

At first glance, doing this doesn't sound like a good idea to me. Among the reasons:

Any method for taking adjacent weekly values and inserting some estimate for the 6 six "missing" daily values will only bias the estimates of day-to-day variance in scores/values as well as increase the serial correlation across a lag from 1 to 6 days. As well, the presumption of a (perfectly) predictable day to day change is unlikely to be realized in actual measurements.

Good luck with your work.

Please, see the attached file...

Alireza Akbari

please write the ODE, not only the discretization You used.

however, for multi-step methods you need to create the starting values using a single-step method for all required nodes. Use a discretization of the same accuracy.

Note that the second order ODE could be written as system of two first order ODE.