Test for Master studs Date: 8/05/2021

Test for Master studs

Date: 8/05/2021

Name: Esirgapov Jamshid

Department: _ Faculty of Mathematics and Informatics,__

University: _ Jizzakh State Pedagogical Institute of Uzbekistan___

Q1: a) Find for the given integrals by using Leibniz rule

Solution:

⁴u(t)dt= ³u(t)dt

b) Convert each of the Volterra integral equations to an equivalent initial value problem

.

Solution:

u(x)=x²+

u^I(x)=2x+

u^II(x)=2+

u^III(x)=

u(0)=0 u^I(x)=0 u^III(x)- u(x)=0

Q2: Convert the following initial value problem to equivalent Volterra integral equation

Solution:

y^II+y^I-2y=2x y(o)=0 y^I(0)=1

y^II=u(x)

= y^I(t) =

y^I(x)-y^I(0)= y^I(x)=1+ 1+

y(x)-y(0)=x+ y(x)=x+

u(x)+1+

u(x)=4x-1+

Q3: a) Solve the following Fredholm integral equations by using the Adomian decomposition method

Solution:

u₀(x)=1+sec²x=1+

u₁(x)

b) Solve the given Fredholm integral equations by using the modified decomposition method

c) Solve the following Fredholm integral equations by using the successive approximations method.

Solution:

Q4: a) Solve the following Fredholm integral equations by using the direct computation method.

Solution:

b) Solve the following Fredholm integral equations by using the successive substitutions method.



Solution: