We present a strong convergence implicit Runge-Kutta method, with four stages, for solution of
initial value problem of ordinary differential equations. Collocation method is used to derive a continuous
scheme; and the continuous scheme evaluated at special points, the Gaussian points of fourth degree Legendre
polynomial, gives us four function evaluations and the Runge-Kutta method for the iteration of the solutions.
Convergent properties of the method are discussed. Experimental problems used to check the quality of the
scheme show that the method is highly efficient, A – stable, has simple structure, converges to exact solution
faster and better than some existing popular methods cited in this paper.
Comparative analysis of x^3+y^3=z^3 and x^2+y^2=z^2 in the Interconnected Sets Vladimir Godovalov
This paper introduces an innovative technique of study z^3-x^3=y^3 on the subject of its insolvability in integers. Technique starts from building the interconnected, third degree sets: A3={a_n│a_n=n^3,n∈N}, B3={b_n│b_n=a_(n+1)-a_n }, C3={c_n│c_n=b_(n+1)-b_n } and P3={6} wherefrom we get a_n and b_n expressed as figurate polynomials of third degree, a new finding in mathematics. This approach and the results allow us to investigate equation z^3-x^3=y in these interconnected sets A3 and B3, where z^3∧x^3∈A3, y∈B3. Further, in conjunction with the new Method of Ratio Comparison of Summands and Pascal’s rule, we finally prove inability of y=y^3. After we test the technique, applying the same approach to z^2-x^2=y where we get family of primitive z^2-x^2=y^2 as well as introduce conception of the basic primitiveness of z^'2-x^'2=y^2 for z^'-x^'=1 and the dependant primitiveness of z^'2-x^'2=y^2 for co-prime x,y,z and z^'-x^'>1.
We present a strong convergence implicit Runge-Kutta method, with four stages, for solution of
initial value problem of ordinary differential equations. Collocation method is used to derive a continuous
scheme; and the continuous scheme evaluated at special points, the Gaussian points of fourth degree Legendre
polynomial, gives us four function evaluations and the Runge-Kutta method for the iteration of the solutions.
Convergent properties of the method are discussed. Experimental problems used to check the quality of the
scheme show that the method is highly efficient, A – stable, has simple structure, converges to exact solution
faster and better than some existing popular methods cited in this paper.
Comparative analysis of x^3+y^3=z^3 and x^2+y^2=z^2 in the Interconnected Sets Vladimir Godovalov
This paper introduces an innovative technique of study z^3-x^3=y^3 on the subject of its insolvability in integers. Technique starts from building the interconnected, third degree sets: A3={a_n│a_n=n^3,n∈N}, B3={b_n│b_n=a_(n+1)-a_n }, C3={c_n│c_n=b_(n+1)-b_n } and P3={6} wherefrom we get a_n and b_n expressed as figurate polynomials of third degree, a new finding in mathematics. This approach and the results allow us to investigate equation z^3-x^3=y in these interconnected sets A3 and B3, where z^3∧x^3∈A3, y∈B3. Further, in conjunction with the new Method of Ratio Comparison of Summands and Pascal’s rule, we finally prove inability of y=y^3. After we test the technique, applying the same approach to z^2-x^2=y where we get family of primitive z^2-x^2=y^2 as well as introduce conception of the basic primitiveness of z^'2-x^'2=y^2 for z^'-x^'=1 and the dependant primitiveness of z^'2-x^'2=y^2 for co-prime x,y,z and z^'-x^'>1.
A Probabilistic Algorithm for Computation of Polynomial Greatest Common with ...mathsjournal
In the earlier work, Knuth present an algorithm to decrease the coefficient growth in the Euclidean algorithm of polynomials called subresultant algorithm. However, the output polynomials may have a small factor which can be removed. Then later, Brown of Bell Telephone Laboratories showed the subresultant in another way by adding a variant called 𝜏 and gave a way to compute the variant. Nevertheless, the way failed to determine every 𝜏 correctly.
In this paper, we will give a probabilistic algorithm to determine the variant 𝜏 correctly in most cases by adding a few steps instead of computing 𝑡(𝑥) when given 𝑓(𝑥) and𝑔(𝑥) ∈ ℤ[𝑥], where 𝑡(𝑥) satisfies that 𝑠(𝑥)𝑓(𝑥) + 𝑡(𝑥)𝑔(𝑥) = 𝑟(𝑥), here 𝑡(𝑥), 𝑠(𝑥) ∈ ℤ[𝑥]
* Plot ordered pairs in a Cartesian coordinate system.
* Graph equations by plotting points.
* Find x-intercepts and y-intercepts.
* Use the distance formula.
* Use the midpoint formula.
The well-known methods' that utilized in the numerical techniques. Several examples illustrated to make the numerical methods that presented more sensible. The Excel program is utilized in this paper.
We disclose a simple and straightforward method of solving single-order linear partial differential equations. The advantage of the method is that it is applicable to any orders and the big disadvantage is that it is restricted to a single order at a time. As it is very easy compared to classical methods, it has didactic value.
Concepts and Applications of the Fundamental Theorem of Line Integrals.pdfJacobBraginsky
A three-part examination of the Fundamental Theorem of Line Integrals. Learn how to use this theorem in multivariable calculus. Simplify the process of solving line integrals using the FTLI.
Partial differentiation, total differentiation, Jacobian, Taylor's expansion, stationary points,maxima & minima (Extreme values),constraint maxima & minima ( Lagrangian multiplier), differentiation of implicit functions.
Module 5 (Part 3-Revised)-Functions and Relations.pdfGaleJean
The Cavite Mutiny of 1872 was an uprising of military personnel of Fort San Felipe, the Spanish arsenal in Cavite, Philippines on January 20, 1872, about 200 Filipino military personnel of Fort San Felipe Arsenal in Cavite, Philippines, staged a mutiny which in a way led to the Philippine Revolution in 1896. The 1872 Cavite Mutiny was precipitated by the removal of long-standing personal benefits to the workers such as tax (tribute) and forced labor exemptions on order from the Governor General Rafael de Izquierdo. Many scholars believe that the Cavite Mutiny of 1872 was the beginning of Filipino nationalism that would eventually lead to the Philippine Revolution of 1896.
Symptoms like intermittent starting and key recognition errors signal potential problems with your Mercedes’ EIS. Use diagnostic steps like error code checks and spare key tests. Professional diagnosis and solutions like EIS replacement ensure safe driving. Consult a qualified technician for accurate diagnosis and repair.
Things to remember while upgrading the brakes of your carjennifermiller8137
Upgrading the brakes of your car? Keep these things in mind before doing so. Additionally, start using an OBD 2 GPS tracker so that you never miss a vehicle maintenance appointment. On top of this, a car GPS tracker will also let you master good driving habits that will let you increase the operational life of your car’s brakes.
A Probabilistic Algorithm for Computation of Polynomial Greatest Common with ...mathsjournal
In the earlier work, Knuth present an algorithm to decrease the coefficient growth in the Euclidean algorithm of polynomials called subresultant algorithm. However, the output polynomials may have a small factor which can be removed. Then later, Brown of Bell Telephone Laboratories showed the subresultant in another way by adding a variant called 𝜏 and gave a way to compute the variant. Nevertheless, the way failed to determine every 𝜏 correctly.
In this paper, we will give a probabilistic algorithm to determine the variant 𝜏 correctly in most cases by adding a few steps instead of computing 𝑡(𝑥) when given 𝑓(𝑥) and𝑔(𝑥) ∈ ℤ[𝑥], where 𝑡(𝑥) satisfies that 𝑠(𝑥)𝑓(𝑥) + 𝑡(𝑥)𝑔(𝑥) = 𝑟(𝑥), here 𝑡(𝑥), 𝑠(𝑥) ∈ ℤ[𝑥]
* Plot ordered pairs in a Cartesian coordinate system.
* Graph equations by plotting points.
* Find x-intercepts and y-intercepts.
* Use the distance formula.
* Use the midpoint formula.
The well-known methods' that utilized in the numerical techniques. Several examples illustrated to make the numerical methods that presented more sensible. The Excel program is utilized in this paper.
We disclose a simple and straightforward method of solving single-order linear partial differential equations. The advantage of the method is that it is applicable to any orders and the big disadvantage is that it is restricted to a single order at a time. As it is very easy compared to classical methods, it has didactic value.
Concepts and Applications of the Fundamental Theorem of Line Integrals.pdfJacobBraginsky
A three-part examination of the Fundamental Theorem of Line Integrals. Learn how to use this theorem in multivariable calculus. Simplify the process of solving line integrals using the FTLI.
Partial differentiation, total differentiation, Jacobian, Taylor's expansion, stationary points,maxima & minima (Extreme values),constraint maxima & minima ( Lagrangian multiplier), differentiation of implicit functions.
Module 5 (Part 3-Revised)-Functions and Relations.pdfGaleJean
The Cavite Mutiny of 1872 was an uprising of military personnel of Fort San Felipe, the Spanish arsenal in Cavite, Philippines on January 20, 1872, about 200 Filipino military personnel of Fort San Felipe Arsenal in Cavite, Philippines, staged a mutiny which in a way led to the Philippine Revolution in 1896. The 1872 Cavite Mutiny was precipitated by the removal of long-standing personal benefits to the workers such as tax (tribute) and forced labor exemptions on order from the Governor General Rafael de Izquierdo. Many scholars believe that the Cavite Mutiny of 1872 was the beginning of Filipino nationalism that would eventually lead to the Philippine Revolution of 1896.
Symptoms like intermittent starting and key recognition errors signal potential problems with your Mercedes’ EIS. Use diagnostic steps like error code checks and spare key tests. Professional diagnosis and solutions like EIS replacement ensure safe driving. Consult a qualified technician for accurate diagnosis and repair.
Things to remember while upgrading the brakes of your carjennifermiller8137
Upgrading the brakes of your car? Keep these things in mind before doing so. Additionally, start using an OBD 2 GPS tracker so that you never miss a vehicle maintenance appointment. On top of this, a car GPS tracker will also let you master good driving habits that will let you increase the operational life of your car’s brakes.
What Could Cause The Headlights On Your Porsche 911 To Stop WorkingLancer Service
Discover why your Porsche 911 headlights might flicker out unexpectedly. From aging bulbs to electrical gremlins and moisture mishaps, we're delving into the reasons behind the blackout. Stay tuned to illuminate the road ahead and ensure your lights shine bright for safer journeys.
Ever been troubled by the blinking sign and didn’t know what to do?
Here’s a handy guide to dashboard symbols so that you’ll never be confused again!
Save them for later and save the trouble!
In this presentation, we have discussed a very important feature of BMW X5 cars… the Comfort Access. Things that can significantly limit its functionality. And things that you can try to restore the functionality of such a convenient feature of your vehicle.
Why Is Your BMW X3 Hood Not Responding To Release CommandsDart Auto
Experiencing difficulty opening your BMW X3's hood? This guide explores potential issues like mechanical obstruction, hood release mechanism failure, electrical problems, and emergency release malfunctions. Troubleshooting tips include basic checks, clearing obstructions, applying pressure, and using the emergency release.
5 Warning Signs Your BMW's Intelligent Battery Sensor Needs AttentionBertini's German Motors
IBS monitors and manages your BMW’s battery performance. If it malfunctions, you will have to deal with an array of electrical issues in your vehicle. Recognize warning signs like dimming headlights, frequent battery replacements, and electrical malfunctions to address potential IBS issues promptly.
Comprehensive program for Agricultural Finance, the Automotive Sector, and Empowerment . We will define the full scope and provide a detailed two-week plan for identifying strategic partners in each area within Limpopo, including target areas.:
1. Agricultural : Supporting Primary and Secondary Agriculture
• Scope: Provide support solutions to enhance agricultural productivity and sustainability.
• Target Areas: Polokwane, Tzaneen, Thohoyandou, Makhado, and Giyani.
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• Scope: Develop collaborations with automotive service providers to improve service quality and business operations.
• Target Areas: Polokwane, Lephalale, Mokopane, Phalaborwa, and Bela-Bela.
3. Empowerment : Focusing on Women Empowerment
• Scope: Provide business support support and training to women-owned businesses, promoting economic inclusion.
• Target Areas: Polokwane, Thohoyandou, Musina, Burgersfort, and Louis Trichardt.
We will also prioritize Industrial Economic Zone areas and their priorities.
Sign up on https://profilesmes.online/welcome/
To be eligible:
1. You must have a registered business and operate in Limpopo
2. Generate revenue
3. Sectors : Agriculture ( primary and secondary) and Automative
Women and Youth are encouraged to apply even if you don't fall in those sectors.
"Trans Failsafe Prog" on your BMW X5 indicates potential transmission issues requiring immediate action. This safety feature activates in response to abnormalities like low fluid levels, leaks, faulty sensors, electrical or mechanical failures, and overheating.
What Are The Immediate Steps To Take When The VW Temperature Light Starts Fla...Import Motorworks
Learn how to respond when the red temperature light flashes in your VW with this presentation. From checking coolant levels to seeking professional help, follow these steps promptly to prevent engine damage and ensure safety on the road.
The Octavia range embodies the design trend of the Škoda brand: a fusion of
aesthetics, safety and practicality. Whether you see the car as a whole or step
closer and explore its unique features, the Octavia range radiates with the
harmony of functionality and emotion
𝘼𝙣𝙩𝙞𝙦𝙪𝙚 𝙋𝙡𝙖𝙨𝙩𝙞𝙘 𝙏𝙧𝙖𝙙𝙚𝙧𝙨 𝙞𝙨 𝙫𝙚𝙧𝙮 𝙛𝙖𝙢𝙤𝙪𝙨 𝙛𝙤𝙧 𝙢𝙖𝙣𝙪𝙛𝙖𝙘𝙩𝙪𝙧𝙞𝙣𝙜 𝙩𝙝𝙚𝙞𝙧 𝙥𝙧𝙤𝙙𝙪𝙘𝙩𝙨. 𝙒𝙚 𝙝𝙖𝙫𝙚 𝙖𝙡𝙡 𝙩𝙝𝙚 𝙥𝙡𝙖𝙨𝙩𝙞𝙘 𝙜𝙧𝙖𝙣𝙪𝙡𝙚𝙨 𝙪𝙨𝙚𝙙 𝙞𝙣 𝙖𝙪𝙩𝙤𝙢𝙤𝙩𝙞𝙫𝙚 𝙖𝙣𝙙 𝙖𝙪𝙩𝙤 𝙥𝙖𝙧𝙩𝙨 𝙖𝙣𝙙 𝙖𝙡𝙡 𝙩𝙝𝙚 𝙛𝙖𝙢𝙤𝙪𝙨 𝙘𝙤𝙢𝙥𝙖𝙣𝙞𝙚𝙨 𝙗𝙪𝙮 𝙩𝙝𝙚 𝙜𝙧𝙖𝙣𝙪𝙡𝙚𝙨 𝙛𝙧𝙤𝙢 𝙪𝙨.
Over the 10 years, we have gained a strong foothold in the market due to our range's high quality, competitive prices, and time-lined delivery schedules.
2. False Position Method
This is the oldest method for finding the real root of a
nonlinear equation 𝑓 𝑥 = 0. It is also known as Regula-
Falsi or method of Chords. In this method, we choose two
points a and b such that 𝑓 𝑎 and 𝑓 𝑏 are of opposite signs,
then a root must lie in between a and b. Now the equation of
the chord joining the two points [𝑎, 𝑓 𝑎 ] and [𝑏, 𝑓 𝑏 ] is
given by-
𝑦−𝑓(𝑎)
𝑥−𝑎
=
𝑓(𝑏)−𝑓(𝑎)
𝑏−𝑎
By taking the point of intersection on x-axis [y=0] as an
approximation to the root we obtain-
2
3. False Position Method (Cont’d)
0−𝑓(𝑎)
𝑥−𝑎
=
𝑓(𝑏)−𝑓(𝑎)
𝑏−𝑎
𝑥 − 𝑎 = −
(𝑏−𝑎) 𝑓(𝑎)
𝑓(𝑏)−𝑓(𝑎)
𝑥 = 𝑎 −
(𝑏−𝑎) 𝑓(𝑎)
𝑓(𝑏)−𝑓(𝑎)
𝑥 =
𝑎𝑓(𝑏)−𝑏𝑓(𝑎)
𝑓(𝑏)−𝑓(𝑎)
Which is the first approximation to the root of 𝑓 𝑥 = 0.
3
4. False Position Method (Cont’d)
4
Fig: Graphical representation of the False Position method.
5. False Position Method (Cont’d)
It can be easily programmed using the following computational
steps:
1. Choose two real numbers a and b such that 𝑓 𝑎 𝑓 𝑏 < 0. [N.B.: a and b must
be closer as much as possible]
2. Set 𝑥𝑟 =
𝑎𝑓(𝑏)−𝑏𝑓(𝑎)
𝑓(𝑏)−𝑓(𝑎)
.
3. (a) If 𝑓 𝑎 𝑓 𝑥𝑟 < 0, the root lies in the interval (𝑎, 𝑥𝑟). Then, set 𝑏 = 𝑥𝑟 and
go to step 2 above.
(b) If 𝑓 𝑎 𝑓 𝑥𝑟 > 0, the root lies in the interval (𝑥𝑟, 𝑏). Then, set
𝑎 = 𝑥𝑟 and go to step 2 above.
(c) If 𝑓 𝑎 𝑓 𝑥𝑟 = 0, it means that 𝑥𝑟 is a root of the equation 𝑓 𝑥 = 0 and
the computation may be terminated.
5
6. False Position Method (Cont’d)
Example: Find a real root of the equation 𝑥3
−2𝑥 − 5 = 0 correct
to three decimal places.
Solution: Given that,
𝑥3
− 2𝑥 − 5 = 0
Let, 𝑓 𝑥 = 𝑥3
−2𝑥 − 5
Taking a=2 and b=3
∴ 𝑓 2 = 23
−2 × 2 − 5 = −1 (< 0)
∴ 𝑓 3 = 33
−2 × 3 − 5 = 16 (> 0)
Hence a root lies between 2 and 3.
6
7. 7
False Position Method (Cont’d)
Step No. a b 𝒙𝒓=
𝒂𝒇(𝒃)−𝒃𝒇(𝒂)
𝒇(𝒃)−𝒇(𝒂)
𝒇 𝒙 = 𝒙𝟑−𝟐𝒙 − 𝟓
1 2 3 2.058823529 −0.390799918
2 2.058823529 3 2.08126366 −0.147204064
3 2.08126366 3 2.089639211 −0.054676498
4 2.089639211 3 2.092739575 −0.020202859
5 2.092739575 3 2.093883709 −0.007450502
6 2.093883709 3 2.094305451 −0.002745669
7 2.094305451 3 2.094460845 −0.001011578
The difference between 𝒙𝟔 and 𝒙𝟕 is 0.000155394. Hence a
root, correct to three decimal places is 2.094.
8. 8
False Position Method(Cont’d)
Self Practise
Example- 2.7, 2.8, 2.9
N.B.: While computing trigonometric ratio, the specified
value must be converted into degree using,
Degree =
180 × 𝑋
3.14
Where, X is in radian unit.
9. Task related to False position Method
1. Write some advantages of false position method.
2. Write some disadvantages of false position method.
9