# Integral Equations

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

solution Solve ${\displaystyle u(x)=f(x)+\int _{0}^{x}k(x-y)u(y)\,dy}$

solution Solve ${\displaystyle u(x)=x+\int _{0}^{x}(x-y)u(y)\,dy}$

solution Formulate an integral equation from the BVP: ${\displaystyle u''+p(x)u'+q(x)u=f(x),\,\,\,x>a}$

solution Solve ${\displaystyle y''(x)=f(x),y(0)=A,y'(0)=B\,}$

solution Approximate ${\displaystyle y(x)=x^{2}+\int _{0}^{1}\sin(xz)y(z)dz\,}$

solution Find lambda: ${\displaystyle y(x)=\lambda \int _{0}^{1}y(t)dt\,}$ if ${\displaystyle y(x)=c\,}$

solution Find lambda: ${\displaystyle y(x)=\lambda \int _{0}^{1}xty(t)dt\,}$ if ${\displaystyle y(x)=x\,}$

solution Find lambda: ${\displaystyle y(x)=\lambda \int _{0}^{1}(x^{2}-z^{2})y(z)dz\,}$

solution Write as an ODE: ${\displaystyle \int _{a}^{b}f(x,y){\sqrt {1+y'^{2}}}\,dx\,}$

solution Solve ${\displaystyle y(x)=\lambda \int _{0}^{1}e^{x+z}y(z)dz,y(x)=e^{x}\,}$

solution Formulate an integral equation from the IVP: ${\displaystyle y''-\lambda y=f(x),x>0,y(0)=1,y'(0)=0\,}$

solution Solve ${\displaystyle f(x)=\lambda \int _{0}^{2}\pi \sin(x+t)y(t)dt\,}$

solution Solve ${\displaystyle f(x)+\lambda \int _{0}^{1}xe^{z}f(z)dz\,}$

solution Find the Euler equation ${\displaystyle J(u)=\int \int _{\mathbb {R} }(y^{2}u_{x}^{2}+y^{2}uy^{2})dxdy\,}$

solution Reduce to a PDE: ${\displaystyle J(u)=\int \int _{\mathbb {R} }dxdy\,}$

solution Transform the BVP to an integral equation: ${\displaystyle {\frac {d^{2}y}{dx^{2}}}+y=x,y(0)=0,y'(1)=0\,}$.

solution Find the value of lambda for which the homogeneous Fredholm integral equation ${\displaystyle y(x)=\lambda \int _{0}^{1}e^{x}e^{t}y(t)dt\,}$ has a nontrivial solution, and find all the solutions.

solution Determine all values of the constants ${\displaystyle a,b,c\,}$ for which the integral equation ${\displaystyle \int _{0}^{1}(1-xt)y(t)dt=ax^{2}+bx+c\,}$ has solutions.

solution Solve: ${\displaystyle g(s)=f(s)+\lambda \int _{0}^{2\pi }\sin(s)\cos(t)g(t)dt\,}$

solution Solve: ${\displaystyle u(x)=\int _{0}^{x}e^{x-y}u(y)dy\,}$

solution Solve: ${\displaystyle \sin(x)=\int _{0}^{x}e^{x-t}u(t)dt\,}$

solution Convert to an integral equation: ${\displaystyle y''+y=\cos(x),y(0)=0,y'(0)=0\,}$

INTEGRAL EQUATIONS BOOKS