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$Solve the wave equation 25 (𝝏^𝟐 𝒖)/𝝏𝒙^𝟐 =(𝝏^𝟐 𝒖)/𝝏𝒕^𝟐 𝒖(𝒙,𝟎)={(𝟐𝟎𝒙 𝟎≤𝒙≤𝟏, 𝟓(𝟓−𝒙) 𝟏≤𝒙≤𝟓) h=1, 𝟎≤𝒕≤𝟏.$

National Science Day-2024

Math E Connect

Math E Connect

National Science Day-2024

Solve for p 𝒑^𝟐+𝟐𝒑𝒚𝒄𝒐𝒕𝒙=𝒚^𝟐

Solve 𝟑𝟐 𝝏𝒖/𝝏𝒕=(𝝏^𝟐 𝒖)/〖𝝏𝒙〗^𝟐 (𝟎, 𝟏) given that h=0.25, upto t=5 𝒖=(𝟎,𝒕)=𝟎=𝒖(𝟏,𝒕), 𝒖=(𝒙,𝟎).

Solve 𝝏𝒖/𝝏𝒕=(𝝏^𝟐 𝒖)/𝝏𝒙^𝟐 (𝟎, 𝟏) at t=0.002 𝒖=(𝟎,𝒕)=𝟎=𝒖(𝟏,𝒕), 𝒖(𝒙,𝟎)={(𝟐𝒙 (𝟎, 𝟎.𝟓) @𝟐(𝟏−𝒙), (𝟎.𝟓, 𝟏)

Solve 𝝏𝒖/𝝏𝒕=(𝝏^𝟐 𝒖)/𝝏𝒙^𝟐, 𝒖=(𝟎,𝒕)=𝟎=𝒖(𝟏,𝒕), 𝒖(𝒙,𝟎)=𝒔𝒊𝒏𝝅𝒙 by taking h=0.2, (𝟎, 𝟎.𝟏) #matheconnect

Find the numerical solution (𝝏^𝟐 𝒖)/𝝏𝒙^𝟐 =𝟐 𝝏𝒖/𝝏𝒕 , 𝒖=(𝟎,𝒕)=𝟎=𝒖(𝟒,𝒕), 𝒖(𝒙,𝟎)=𝒙(𝟒−𝒙) ,h=1,t=4 steps.

Numerical Solution of the one dimensional Heat equation & working procedure.

Show that: 𝒊)𝑷_𝟐(𝒄𝒐𝒔𝜽)=1/4 (𝟏+𝟑𝒄𝒐𝒔𝜽), 𝒊𝒊) 𝑷_𝟑 (𝒄𝒐𝒔𝜽)=𝟏/𝟖 (𝟑𝒄𝒐𝒔𝜽+𝟓𝒄𝒐𝒔𝟑𝜽).

Express 𝒇(𝒙)=𝒙^𝟒−𝟑𝒙^𝟑−𝒙^𝟐+𝟓𝒙 in terms of Legendre polynomial.

Express 𝟐𝒙^𝟑−𝒙^𝟐−𝟑𝒙+𝟐 in terms of Legendre polynomial

If 𝒙^𝟑+𝟐𝒙^𝟐−𝒙+𝟏=𝒂𝑷_𝟎 (𝒙)+𝐛_𝑷 𝟏( 𝒙)+c𝑷 _𝟐 (𝒙) +d𝑷_𝟑 (𝒙) .Find the value of a, b, c, d.

Express 𝒙^𝟑+𝟐𝒙^𝟐−𝟒𝒙+𝟓 in terms of Legendre polynomial

Rodrigues Formula & Results.

Series Solution of Legendre s Differential Equation

Orthogonality of Bessel s Functions

Prove that : 𝑱((−𝟏)⁄𝟐) (𝒙)=√(𝟐/𝝅𝒙) 𝒄𝒐𝒔𝒙.

Prove that : 𝑱(𝟏⁄𝟐) (𝒙)=√(𝟐/𝝅𝒙) 𝒔𝒊𝒏𝒙.

Bessel's Functions: Series solution of Bessel's Differential Equation Leading to Bessel's Function.

Solve the wave equation (𝝏^𝟐 𝒖)/〖𝝏𝒙〗^𝟐 =(𝝏^𝟐 𝒖)/〖𝝏𝒕〗^𝟐 𝒖(𝒙,𝟎)=𝒔𝒊𝒏𝝅𝒙 (0,1), h=0.2, k=0.2 at 𝒕=𝟏.

Solve the wave equation 25 (𝝏^𝟐 𝒖)/𝝏𝒙^𝟐 =(𝝏^𝟐 𝒖)/𝝏𝒕^𝟐 𝒖(𝒙,𝟎)={(𝟐𝟎𝒙 𝟎≤𝒙≤𝟏, 𝟓(𝟓−𝒙) 𝟏≤𝒙≤𝟓) h=1, 𝟎≤𝒕≤𝟏.

Solve the Wave equation (𝝏^𝟐 𝒖)/(𝝏𝒙^𝟐) =𝟎.𝟎𝟔𝟓 (𝝏^𝟐 𝒖)/𝝏𝒕^𝟐 , 𝒖(𝒙,𝟎)=𝒙^𝟐 (𝒙−𝟓), h=1, k=0.5 ,4 steps.

Solve the wave equation (𝝏^𝟐 𝒖)/〖𝝏𝒕〗^𝟐 =𝟒 (𝝏^𝟐 𝒖)/〖𝝏𝒙〗^𝟐 𝒖(𝒙,𝟎)=𝒙(𝟒−𝒙) , h=1, k=0.5 upto 4 steps.

Working procedure for one dimensional wave equation

21MAT31: Numerical solution of the one dimensional wave equation.

21MAT31:Numerical Method PDE, Classification of PDE of second order & Finite difference approximate.

# Partial Derivatives || # Definition || #Symbols

#Evaluate || lim_𝒙→𝟎⁡(𝒔𝒊𝒏𝒙𝒙^𝟏⁄𝒙^𝟐 ) || 21MAT11 || Module : 2 || Differential Calculus ||BMATE101

Evaluate || lim_𝒙→𝟎⁡(𝒕𝒂𝒏𝒙𝒙^𝟏⁄𝒙 ) || 21MAT11 || Module : 2 || Differential Calculus ||

Evaluate || lim┬(𝒙→𝟎)⁡[((𝒂^𝒙+𝒃^𝒙+𝒄^𝒙+𝒅^𝒙 ))/𝟒]^(𝟏⁄𝒙) || 18MAT11 || Differential Calculus

Evaluate || lim_𝑥→𝜋/2⁡(𝒔𝒊𝒏𝒙^𝒕𝒂𝒏𝒙 )|| 18MAT11 || Differential Calculus

Evaluate || lim┬(𝒙→𝟎)⁡ ([𝒄𝒐𝒔𝒙]^(𝟏⁄𝒙^𝟐 )) || Differential Calculus || #Math E Connect

Evaluate || lim_(𝒙→𝟏)⁡(𝒙^(𝟏⁄(𝟏−𝒙)) )|| Differential Calculus || #Math E Connect||

How to evaluate (𝟎)^𝟎 , (∞)^𝟎 (𝟏)^∞ || Differential Calculus. ||

Evaluate || lim_(𝒙→𝟎)⁡[𝟏/𝒙^𝟐 −𝟏/((𝒔𝒊𝒏)^𝟐 𝒙)] || Differential Calculus ||

Evaluate || lim_(𝒙→𝟎)⁡[𝟏𝒙^𝟐 −(𝒄𝒐𝒕)^𝟐 𝒙] || Differential Calculus ||

Evaluate || lim_(x→0)⁡[(tanx-x)/(x^2 tanx)] ||..

Evaluate || lim_(x→a) log⁡[2-x/a]cot(x-a) ||.

Evaluate || lim_(x→0)⁡[1/x-(log(1+x))/x^2 ].

Evaluate the ( lim)_(x→0) ⁡[(1-cosx)/(xlog(1+x))] || Differential Calculus ||

|| Evaluate || (lim⁡ )_(x→1)⁡[ x/(x-1)-1/logx ] || Differential Calculus ||

Evaluate || lim 𝒙→𝟎⁡〖𝒍𝒐𝒈𝒙 / 𝒄𝒐𝒔𝒆𝒄𝒙〗|| Differential Calculus ||

#18 Problem#4||PDE by direct integration||(𝝏^𝟐 𝒛)/𝝏𝒙𝝏𝒚=𝒔𝒊𝒏𝒙𝒔𝒊𝒏𝒚,||𝝏𝒛/𝝏𝒚=−𝟐𝒔𝒊𝒏𝒚, 𝒙=𝟎 and 𝒛=𝟎 𝒚=𝝅/𝟐||

#17||Problem#3||Solution of non-homogenous PDE by direct integration (𝝏^𝟐 𝒛)/(𝝏𝒙^𝟐 𝝏𝒚)=𝒄𝒐𝒔(𝟐𝒙+𝟑𝒚)||

#16 || Problem#2 ||Solution of non-homogenous PDE by direct integration Solve: (𝝏^𝟐 𝒛)/𝝏𝒙𝝏𝒚=𝒙/𝒚+𝒂 ||

#15 || Problem#1|| Solution of non-homogenous PDE by direct integration || Solve:(𝝏^𝟐 𝒖)/𝝏𝒙^𝟐 =𝒙+𝒚||

#14 ||Problem#2 ||Formation of partial differential equations|| Form a PDE from𝒇(𝒙^𝟐+𝒚^𝟐, 𝒛−𝒙𝒚) ||

#13 || Problem#1 || Formation of PDE || Form the PDE from∅(𝒙+𝒚+𝒛,𝒙^𝟐−𝒚^𝟐−𝒛^𝟐) || 18MAT21||

#12 || Formation of PDE by from ∅(𝒖,𝒗)=𝟎 where u v are functions of x, y, z. || Steps||

#30 || Solution of Lagrange’s linear Partial differential equation || Solve: (𝒚^𝟐+𝒙^𝟐)𝒑+𝒙(𝒚𝒒−𝒛)=𝟎 ||

#29 || Solution of Lagrange’s linear Partial differential equation || Solve: 𝒙𝒑−𝒚𝒒=𝒚^𝟐−𝒙^𝟐|| 18MAT21

#28 ||Solution of Lagrange’s linear Partial differential equation ||Solve: 𝒑𝒚𝒛+𝒒𝒛𝒙=𝒙𝒚|| 18MAT21||

#27 || Problem on Solution of Lagrange’s linear PDE|| Solve: (𝒚^𝟐 𝒛)/𝒙 𝒑+𝒙𝒛𝒒=𝒚^𝟐|| 18MAT21|| #PDE||

#26 || Solution of Lagrange’s linear Partial differential equation || Working Rule ||18MAT21||#PDE||

#24 || Solution of one dimensional Heat equation (𝝏^𝟐 𝒖)/𝝏𝒕^𝟐 =𝒄^𝟐 (𝝏^𝟐 𝒖)/𝝏𝒙^𝟐 || 18MAT21 || #PDE.

#25 || Solution of one dimensional Wave equation 𝝏𝒖/𝝏𝒕=𝒄^𝟐 (𝝏^𝟐 𝒖)/〖𝝏𝒙〗^𝟐 || 18MAT21 || #PDE

#21 || Problem#7 || (𝝏^𝟐 𝒛)/〖𝝏𝒙〗^𝟐 =𝒂^𝟐 𝒛, given that when 𝒙=𝟎 and 𝒛=𝟎 and 𝝏𝒛/𝝏𝒙=𝒂𝒔𝒊𝒏𝒚|| 18MAT21 ||

#20 || Problem#6 || (𝝏^𝟐 𝒛)/〖𝝏𝒙〗^𝟐 +𝒛=𝟎, given that when 𝒙=𝟎 and 𝒛=𝒆^𝒚 and 𝝏𝒛/𝝏𝒙=𝟏 ||18MAT21||#PDE

#19 || Problem#5 || (𝝏^𝟐 𝒛)/𝝏𝒙^𝟐 =𝒙𝒚, subject to the condition 𝝏𝒛/𝝏𝒙=𝒍𝒐𝒈(𝟏+𝒚) when 𝒙=𝟏 and 𝒛=𝟎||

#11 || Problem#6 || PDE by eliminating the arbitrary function of the equation𝒛=𝒚𝒇(𝒙)+𝒙∅(𝒚) ||

#10 || Problem#5 || PDE by eliminating the arbitrary function of the equation𝒛=𝒆^(𝒂𝒙+𝒃𝒚) 𝒇(𝒂𝒙−𝒃𝒚) ||

#9 || Problem#5 || PDE by eliminating the arbitrary function of the equation 𝒛=𝒙+𝒚+𝒇(𝒙𝒚) || 18MAT21

#8 || Problem#4 || Form a PDE by eliminating the arbitrary function of the equation𝒛=𝒆^𝒚 𝒇(𝒙+𝒚) ||

#7 || Problem#3 | Form a PDE by eliminating the arbitrary function of the equation 𝒛=𝒇(𝒙^𝟐+𝒚^𝟐) ||

#6 || Problem#2 || PDE by eliminating the arbitrary function of the equation𝒛=𝒇(𝒙+𝒂𝒕)+𝒈(𝒙−𝒂𝒕) ||

#5 || Problem#1|| PDE by eliminating the arbitrary function of the equation. 𝒛=(𝒙+𝒚)∅(𝒙^𝟐−𝒚^𝟐) ||

#4 || Problem#3 || PDE || Eliminating the arbitrary constant || 𝒙^𝟐/𝒂^𝟐 +𝒚^𝟐/𝒃^𝟐 +𝒛^𝟐/𝒄^𝟐 =𝟏 ||

#3 ||Problem #3 || PDE ||eliminating the arbitrary constant from the equation𝟐𝒛=𝒙^𝟐/𝒂^𝟐 +𝒚^𝟐/𝒃^𝟐 |

#2 || Problem#1||Formation of partial differential equations by eliminating the arbitrary constant |

#1 || Introduction || Partial Differential Equations || 18MAT21 || #PDE || #Shafiqahmedyellur

#21 Problem#2 || ∫(𝒔𝒊𝒏𝝅𝒛^𝟐+𝒄𝒐𝒔𝝅𝒛^𝟐)/((𝒛−𝟏)^𝟐 (𝒛−𝟐)) 𝒅𝒛|| c:|𝒛|=𝟑 || c:|𝒛|=𝟏/𝟐 || c: |𝒛|=𝟑/𝟐||18MAT41

#20 || Problem#1|| Cauchy’s integral formula|| ∮(𝒔𝒊𝒏𝝅𝒛^𝟐+𝒄𝒐𝒔𝝅𝒛^𝟐)/((𝒛−𝟏)(𝒛−𝟐)) 𝒅𝒛 || c: |𝒛|=𝟑 ||

#18 || Problem#5|| Cauchy’s theorem || ∫(𝒁^𝟐−𝒁+𝟏)/〖(𝒁−𝟐)^𝟑 𝒅𝒛 over 𝐜: |𝒛|=𝟑 || 18MAT41||

#17 || Problem#4|| Cauchy’s theorem || ∫𝒆^𝟐𝒛/((𝒁+𝟏)^𝟐 (𝒛−𝟐)) 𝒅𝒛 ,c is the circle |𝒛|=𝟑 ||18MAT41 ||

#16 || Problem#4|| Cauchy’s theorem || ∫𝒆^𝒛/((𝒛+𝟏)(𝒛−𝟐)) 𝒅𝒛 || c: |𝒛|=𝟑|| 18MAT41||

#15 || Problem#3||Cauchy’s theorem ||∫𝒅𝒛/(𝒛^𝟐−𝟒) || 𝒄:|𝒛|=𝟏 || 𝒄:|𝒛|=𝟑 || 𝒄: |𝒛+𝟐|=𝟏 || 18MAT41

#14 ||Problem#2|| Cauchy’s theorem || ∫𝒆^𝒛/(𝒛−𝒊𝝅) 𝒅𝒛 || c:|𝒛|=𝟐𝝅 || c:|𝒛|=𝝅/𝟐|| 18MAT41 ||

#13 || Problem#1 || Cauchy’s theorem || ∫(𝒛^𝟐−𝒛+𝟏)/(𝒛−𝟏) 𝒅𝒛 || c: |𝒛|=𝟏 || c:|𝒛|=𝟏/𝟐|| 18MAT41 ||