Sentence examples for kernel of an integral from inspiring English sources

Proof: From (9), we get that r x (t, τ) is the kernel of an integral operator of L2 into L2, which is linear, self-adjoint, nonnegative-definite, and compact.

According to Mercer's theorem of functional analysis, if the kernel function is a continuous kernel of a positive integral operator, there exists a map, Φ, into a dot product space, F, such that the formula holds.

Another kind of fractional derivative that appears in the literature is the fractional derivative due to Hadamard, introduced in 1892 [12], which differs from the Riemann-Liouville and Caputo derivatives in the sense that the kernel of the integral contains a logarithmic function of an arbitrary exponent.

This fractional derivative differs from the other ones in the sense that the kernel of the integral (in the definition of the Hadamard derivative) contains a logarithmic function of an arbitrary exponent.

Here (F x, y)) is a kernel of integral equation (2.2), where x is a parameter, (F x, y)) is a known function and (K x, y)) is an unknown function, as functions of y.

The Hadamard fractional derivative, introduced by Hadamard in 1892 [12], differs from the Riemann-Liouville and Caputo derivatives in the sense that the kernel of the integral (in the definition of Hadamard derivative) contains a logarithmic function of arbitrary exponent.

Besides these fractional derivatives, another kind of fractional derivatives found in the literature is the fractional derivative due to Hadamard, introduced in 1892 [14], which differs from the aforementioned derivatives in the sense that the kernel of the integral in the definition of Hadamard derivative contains a logarithmic function of arbitrary exponent.

Another kind of fractional derivatives that appears side by side to Riemann-Liouville and Caputo derivatives in the literature is the fractional derivative due to Hadamard introduced in 1892 [22], which differs from the preceding ones in the sense that the kernel of the integral (in the definition of Hadamard derivative) contains a logarithmic function of arbitrary exponent.

This derivative differs from the aforementioned derivatives in the sense that the kernel of the integral in the definition of the Hadamard derivative contains a logarithmic function of arbitrary exponent.

where is the kernel of the integral equation.

where. is the heat kernel of the integral representation T t f.