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Minor formatting (Bracket scaling and splitting of formula
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Daniele Tampieri
Daniele Tampieri
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\begin{equation} \begin{split} &\frac{\partial}{\partial t}P(x, t)=\\ &\sum\limits_{i<j}^{n}a_{i,j}\,\frac{x_i-x_j}{1-c_i-c_j}\,(c_i\,\frac{\partial P}{\partial x_i}- c_j\,\frac{\partial P}{\partial x_j}) +\frac 12\sum\limits_{i<j}^{n}a_{i,j}\,(\frac{x_i-x_j}{1-c_i-c_j})^2\,(c_i^2\,\frac{\partial^2 P}{\partial x_i^2}+ c_j^2\,\frac{\partial^2 P}{\partial x_j^2}) \\&-\sum\limits_{i<j}^{n}a_{i,j}\,\frac{c_i\,c_j}{(1-c_i-c_j)^2}\,(x_i-x_j)^2\,\frac{\partial^2 P}{\partial x_i\partial x_j} \end{split} \end{equation}$$ \begin{split} \frac{\partial}{\partial t}P(x, t)& =\sum\limits_{i<j}^{n}a_{i,j}\,\frac{x_i-x_j}{1-c_i-c_j}\,\bigg(c_i\frac{\partial P}{\partial x_i} - c_j\frac{\partial P}{\partial x_j}\bigg)\\ &\qquad+\frac 12\sum\limits_{i<j}^{n}a_{i,j}\bigg(\frac{x_i-x_j}{1-c_i-c_j}\bigg)^2\bigg(c_i^2\,\frac{\partial^2 P}{\partial x_i^2}+ c_j^2\,\frac{\partial^2 P}{\partial x_j^2}\bigg)\\ &\qquad\qquad -\sum\limits_{i<j}^{n}a_{i,j}\,\frac{c_i\,c_j}{(1-c_i-c_j)^2}\,(x_i-x_j)^2\,\frac{\partial^2 P}{\partial x_i\partial x_j} \end{split} $$

I have been trying to solve the above PDE, with different methods such as Fourier Transform and method of separation of variables, but was not able to find a solution. What is the best way to find a general solution to the above PDE?

\begin{equation} \begin{split} &\frac{\partial}{\partial t}P(x, t)=\\ &\sum\limits_{i<j}^{n}a_{i,j}\,\frac{x_i-x_j}{1-c_i-c_j}\,(c_i\,\frac{\partial P}{\partial x_i}- c_j\,\frac{\partial P}{\partial x_j}) +\frac 12\sum\limits_{i<j}^{n}a_{i,j}\,(\frac{x_i-x_j}{1-c_i-c_j})^2\,(c_i^2\,\frac{\partial^2 P}{\partial x_i^2}+ c_j^2\,\frac{\partial^2 P}{\partial x_j^2}) \\&-\sum\limits_{i<j}^{n}a_{i,j}\,\frac{c_i\,c_j}{(1-c_i-c_j)^2}\,(x_i-x_j)^2\,\frac{\partial^2 P}{\partial x_i\partial x_j} \end{split} \end{equation}

I have been trying to solve the above PDE, with different methods such as Fourier Transform and method of separation of variables, but was not able to find a solution. What is the best way to find a general solution to the above PDE?

$$ \begin{split} \frac{\partial}{\partial t}P(x, t)& =\sum\limits_{i<j}^{n}a_{i,j}\,\frac{x_i-x_j}{1-c_i-c_j}\,\bigg(c_i\frac{\partial P}{\partial x_i} - c_j\frac{\partial P}{\partial x_j}\bigg)\\ &\qquad+\frac 12\sum\limits_{i<j}^{n}a_{i,j}\bigg(\frac{x_i-x_j}{1-c_i-c_j}\bigg)^2\bigg(c_i^2\,\frac{\partial^2 P}{\partial x_i^2}+ c_j^2\,\frac{\partial^2 P}{\partial x_j^2}\bigg)\\ &\qquad\qquad -\sum\limits_{i<j}^{n}a_{i,j}\,\frac{c_i\,c_j}{(1-c_i-c_j)^2}\,(x_i-x_j)^2\,\frac{\partial^2 P}{\partial x_i\partial x_j} \end{split} $$

I have been trying to solve the above PDE, with different methods such as Fourier Transform and method of separation of variables, but was not able to find a solution. What is the best way to find a general solution to the above PDE?

added 8 characters in body
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Dan
Dan
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\begin{equation} \begin{split} &\frac{\partial}{\partial t}P(x, t)=\\ &\sum\limits_{i<j}^{n}a_{i,j}\,\frac{x_i-x_j}{1-c_i-c_j}\,(c_i\,\frac{\partial P}{\partial x_i}- c_j\,\frac{\partial P}{\partial x_j}) +\sum\limits_{i<j}^{n}a_{i,j}\,(\frac{x_i-x_j}{1-c_i-c_j})^2\,(c_i^2\,\frac{\partial^2 P}{\partial x_i^2}+ c_j^2\,\frac{\partial^2 P}{\partial x_j^2}) \\&-\sum\limits_{i<j}^{n}a_{i,j}\,\frac{c_i\,c_j}{(1-c_i-c_j)^2}\,(x_i-x_j)^2\,\frac{\partial^2 P}{\partial x_i\partial x_j} \end{split} \end{equation}\begin{equation} \begin{split} &\frac{\partial}{\partial t}P(x, t)=\\ &\sum\limits_{i<j}^{n}a_{i,j}\,\frac{x_i-x_j}{1-c_i-c_j}\,(c_i\,\frac{\partial P}{\partial x_i}- c_j\,\frac{\partial P}{\partial x_j}) +\frac 12\sum\limits_{i<j}^{n}a_{i,j}\,(\frac{x_i-x_j}{1-c_i-c_j})^2\,(c_i^2\,\frac{\partial^2 P}{\partial x_i^2}+ c_j^2\,\frac{\partial^2 P}{\partial x_j^2}) \\&-\sum\limits_{i<j}^{n}a_{i,j}\,\frac{c_i\,c_j}{(1-c_i-c_j)^2}\,(x_i-x_j)^2\,\frac{\partial^2 P}{\partial x_i\partial x_j} \end{split} \end{equation}

I have been trying to solve the above PDE, with different methods such as Fourier Transform and method of separation of variables, but was not able to find a solution. What is the best way to find a general solution to the above PDE?

\begin{equation} \begin{split} &\frac{\partial}{\partial t}P(x, t)=\\ &\sum\limits_{i<j}^{n}a_{i,j}\,\frac{x_i-x_j}{1-c_i-c_j}\,(c_i\,\frac{\partial P}{\partial x_i}- c_j\,\frac{\partial P}{\partial x_j}) +\sum\limits_{i<j}^{n}a_{i,j}\,(\frac{x_i-x_j}{1-c_i-c_j})^2\,(c_i^2\,\frac{\partial^2 P}{\partial x_i^2}+ c_j^2\,\frac{\partial^2 P}{\partial x_j^2}) \\&-\sum\limits_{i<j}^{n}a_{i,j}\,\frac{c_i\,c_j}{(1-c_i-c_j)^2}\,(x_i-x_j)^2\,\frac{\partial^2 P}{\partial x_i\partial x_j} \end{split} \end{equation}

I have been trying to solve the above PDE, with different methods such as Fourier Transform and method of separation of variables, but was not able to find a solution. What is the best way to find a general solution to the above PDE?

\begin{equation} \begin{split} &\frac{\partial}{\partial t}P(x, t)=\\ &\sum\limits_{i<j}^{n}a_{i,j}\,\frac{x_i-x_j}{1-c_i-c_j}\,(c_i\,\frac{\partial P}{\partial x_i}- c_j\,\frac{\partial P}{\partial x_j}) +\frac 12\sum\limits_{i<j}^{n}a_{i,j}\,(\frac{x_i-x_j}{1-c_i-c_j})^2\,(c_i^2\,\frac{\partial^2 P}{\partial x_i^2}+ c_j^2\,\frac{\partial^2 P}{\partial x_j^2}) \\&-\sum\limits_{i<j}^{n}a_{i,j}\,\frac{c_i\,c_j}{(1-c_i-c_j)^2}\,(x_i-x_j)^2\,\frac{\partial^2 P}{\partial x_i\partial x_j} \end{split} \end{equation}

I have been trying to solve the above PDE, with different methods such as Fourier Transform and method of separation of variables, but was not able to find a solution. What is the best way to find a general solution to the above PDE?

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Dan
Dan
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General solution to a n-dimensional partial differential equation

\begin{equation} \begin{split} &\frac{\partial}{\partial t}P(x, t)=\\ &\sum\limits_{i<j}^{n}a_{i,j}\,\frac{x_i-x_j}{1-c_i-c_j}\,(c_i\,\frac{\partial P}{\partial x_i}- c_j\,\frac{\partial P}{\partial x_j}) +\sum\limits_{i<j}^{n}a_{i,j}\,(\frac{x_i-x_j}{1-c_i-c_j})^2\,(c_i^2\,\frac{\partial^2 P}{\partial x_i^2}+ c_j^2\,\frac{\partial^2 P}{\partial x_j^2}) \\&-\sum\limits_{i<j}^{n}a_{i,j}\,\frac{c_i\,c_j}{(1-c_i-c_j)^2}\,(x_i-x_j)^2\,\frac{\partial^2 P}{\partial x_i\partial x_j} \end{split} \end{equation}

I have been trying to solve the above PDE, with different methods such as Fourier Transform and method of separation of variables, but was not able to find a solution. What is the best way to find a general solution to the above PDE?