Linear Integral Equations

Fredholm and Volterra Type, Separable Kernels, Resolvent Kernel

• 88Integral equations — a separable kernel with no real eigenvaluesIntegral equations — a separable kernel with no real eigenvaluesMSET June 2025Fredholm and Volterra Type, Separable Kernels, Resolvent KernelMedium
• 89Integral equations — Abel's equationIntegral equations — Abel's equationMSET June 2025Fredholm and Volterra Type, Separable Kernels, Resolvent KernelMedium
• 90Integral equations — BVP to Fredholm equationIntegral equations — BVP to Fredholm equationMSET June 2025Fredholm and Volterra Type, Separable Kernels, Resolvent KernelMedium
• 91Integral equations — eigenvalues of cos(t+s) on [0,pi]Integral equations — eigenvalues of $\cos (t+s)$ on $[0,\pi ]$MSET June 2025Fredholm and Volterra Type, Separable Kernels, Resolvent KernelMedium
• 92Integral equations — resolvent of a^t-sIntegral equations — resolvent of $a^{t-s}$MSET June 2025Fredholm and Volterra Type, Separable Kernels, Resolvent KernelMedium
• 93Integral equations — a variable lower limit makes it Volterra, second kindIntegral equations — a variable lower limit makes it Volterra, second kindMSET June 2025Fredholm and Volterra Type, Separable Kernels, Resolvent KernelEasy
• 88Classifying an integral equation — fixed limits, unknown outside the integralClassifying an integral equation — fixed limits, unknown outside the integralMSET April 2024Fredholm and Volterra Type, Separable Kernels, Resolvent KernelMedium
• 89A separable-kernel eigenvalue problemA separable-kernel eigenvalue problemMSET April 2024Fredholm and Volterra Type, Separable Kernels, Resolvent KernelEasy
• 93Eigenvalues from a separable trigonometric kernelEigenvalues from a separable trigonometric kernelMSET April 2024Fredholm and Volterra Type, Separable Kernels, Resolvent KernelMedium
• 88A separable-kernel eigenfunction shapeA separable-kernel eigenfunction shapeMSET March 2023Fredholm and Volterra Type, Separable Kernels, Resolvent KernelEasy
• 93Eigenvalues of a degenerate (separable) Fredholm kernelEigenvalues of a degenerate (separable) Fredholm kernelMSET March 2023Fredholm and Volterra Type, Separable Kernels, Resolvent KernelMedium
• 88A symmetric-kernel operator is self-adjoint, with real "diagonal" inner productsA symmetric-kernel operator is self-adjoint, with real "diagonal" inner productsMSET September 2021Fredholm and Volterra Type, Separable Kernels, Resolvent KernelMedium
• 94Recognising a Fourier cosine transform pairRecognising a Fourier cosine transform pairMSET September 2021Fredholm and Volterra Type, Separable Kernels, Resolvent KernelEasy
• 86Non-vanishing Fredholm determinant forces the trivial solution aloneNon-vanishing Fredholm determinant forces the trivial solution aloneMSET December 2020Fredholm and Volterra Type, Separable Kernels, Resolvent KernelEasy
• 87Solving a separable-kernel eigenvalue problem for k(t,s)=e^t+sSolving a separable-kernel eigenvalue problem for $k(t,s)=e^{t+s}$MSET December 2020Fredholm and Volterra Type, Separable Kernels, Resolvent KernelEasy
• 88Solving a convolution-type Volterra equation of the first kind via Laplace transformsSolving a convolution-type Volterra equation of the first kind via Laplace transformsMSET December 2020Fredholm and Volterra Type, Separable Kernels, Resolvent KernelEasy
• 87Building up the iterated kernel for k(t,s)=e^t-sBuilding up the iterated kernel for $k(t,s)=e^{t-s}$MSET June 2019Fredholm and Volterra Type, Separable Kernels, Resolvent KernelEasy
• 88Solving a convolution-kernel Volterra equation via Laplace transformsSolving a convolution-kernel Volterra equation via Laplace transformsMSET June 2019Fredholm and Volterra Type, Separable Kernels, Resolvent KernelEasy
• 62Fixed limits and x only under the integral: Fredholm, first kindFixed limits and $x$ only under the integral: Fredholm, first kindMSET January 2018 Paper IIIFredholm and Volterra Type, Separable Kernels, Resolvent KernelEasy
• 63Differentiate the Volterra equation twice: x"-x=cos tDifferentiate the Volterra equation twice: $x"-x=\cos t$MSET January 2018 Paper IIIFredholm and Volterra Type, Separable Kernels, Resolvent KernelMedium
• 64Abel's equation: the kernel (t-s)^-1/2 inverts to x=2sqrt tAbel's equation: the kernel $(t-s{)}^{-1/2}$ inverts to $x=2\sqrt{t}$MSET January 2018 Paper IIIFredholm and Volterra Type, Separable Kernels, Resolvent KernelHard
• 62Variable upper limit, x outside the integral, kernel depending on t-sVariable upper limit, $x$ outside the integral, kernel depending on $t-s$MSET April 2017 Paper IIIFredholm and Volterra Type, Separable Kernels, Resolvent KernelEasy
• 63With u=y", the BVP becomes u=2x-1+int0^1k1(x,t)u(t),dtWith $u=y"$, the BVP becomes $u=2x-1+{\int }_0^1{k}_1(x,t)u(t)dt$MSET April 2017 Paper IIIFredholm and Volterra Type, Separable Kernels, Resolvent KernelHard
• 64Substitute x=e^t^2w: then w=1+int0^tw, so w=e^tSubstitute $x=e^{t^2}w$: then $w=1+{\int }_0^{t}w$, so $w=e^t$MSET April 2017 Paper IIIFredholm and Volterra Type, Separable Kernels, Resolvent KernelMedium