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A natural extension of the Riemann p-differential equation given by (d^2w)/(dx^2)+(gamma/x+delta/(x-1)+epsilon/(x-a))(dw)/(dx)+(alphabetax-q)/(x(x-1)(x-a))w=0 where ...

The partial differential equation 3/4U_y+W_x=0, (1) where W_y+U_t-1/4U_(xxx)+3/2UU_x=0 (2) (Krichever and Novikov 1980; Novikov 1999). Zwillinger (1997, p. 131) and Calogero ...

A natural equation is an equation which specifies a curve independent of any choice of coordinates or parameterization. The study of natural equations began with the ...

A polynomial given by Phi_n(x)=product_(k=1)^n^'(x-zeta_k), (1) where zeta_k are the roots of unity in C given by zeta_k=e^(2piik/n) (2) and k runs over integers relatively ...

A special case of the quadratic Diophantine equation having the form x^2-Dy^2=1, (1) where D>0 is a nonsquare natural number (Dickson 2005). The equation x^2-Dy^2=+/-4 (2) ...

The second-order ordinary differential equation y^('')+(y^')/x+(1-(nu^2)/(x^2))y=(x-nu)/(pix^2)sin(pinu) whose solutions are Anger functions.

A method which can be used to solve some classes of integral equations and is especially useful in implementing certain types of data inversion. It has been applied to invert ...

A Bäcklund transformation allows additional solutions to a nonlinear partial differential equations to be found if one particular solution is already known.

There are three types of boundary conditions commonly encountered in the solution of partial differential equations: 1. Dirichlet boundary conditions specify the value of the ...

A symmetry of a differential equation is a transformation that keeps its family of solutions invariant. Symmetry analysis can be used to solve some ordinary and partial ...