# Solving lengthy higher order pde using finite difference

I am attempting to solve a very massive pde (latex expression given below).

I have done some work with finite difference before for relatively simple equations (like heat diffusion or the wave equation). However, most examples are done for only second order equations, and those are relatively short.

For example, I have terms like

$\frac{4 L^2 \chi ^{(0,2)}(x,t)}{x^2}-1.31754 x^8 \left(\left(-0.573921 \left(c^2-1\right) x^2+x-1\right) \chi ^{(4,0)}(x,t)-1.14784 \left(c^2-1\right) \chi ^{(2,0)}(x,t)+2 \left(1-1.14784 \left(c^2-1\right) x\right) \chi ^{(3,0)}(x,t)\right)==0$

Obviously, when dealing with many higher order terms like this, coding a solution by hand quickly grows unyielding. What I am curious about is if there is a generally faster method for writing up code for higher order PDEs, or if possible a Mathematica package that could greatly help simplify things.

I am not looking for a full solution by any means (especially because all the boundary conditions haven't been determined yet), but was more curious as to a method that could be more or less readily generalised to length higher order expressions without a massive overhaul of code.

$\frac{1}{2} (1.31754 \left(-1.14784 \left(c^2-1\right) \chi ^{(2,0)}(x,t)+2 \left(1-1.14784 \left(c^2-1\right) x\right) \chi ^{(3,0)}(x,t)+\left(-0.573921 \left(c^2-1\right) x^2+x-1\right) \chi ^{(4,0)}(x,t)\right) x^8+21.0807 \left(\left(1-1.14784 \left(c^2-1\right) x\right) \chi ^{(2,0)}(x,t)+\left(-0.573921 \left(c^2-1\right) x^2+x-1\right) \chi ^{(3,0)}(x,t)\right) x^7+73.7823 \left(-0.573921 \left(c^2-1\right) x^2+x-1\right) \chi ^{(2,0)}(x,t) x^6-\frac{0.869577 (1.14784 x+1) \chi ^{(2,1)}(x,t) x^5}{\sqrt{0.573921 x^2+x-1}}-1.73915 \sqrt{0.573921 x^2+x-1} \chi ^{(3,1)}(x,t) x^5-\frac{0.434789 \left((3.0303 (0.757576 (x-1)-0.757576 x)-1.) \chi ^{(1,1)}(x,t)+4 \left(0.573921 x^2+x-1\right) \left((1.14784 x+1) \chi ^{(2,1)}(x,t)+\left(0.573921 x^2+x-1\right) \chi ^{(3,1)}(x,t)\right)\right) x^5}{\left(0.573921 x^2+x-1\right)^{3/2}}+\frac{0.573921 \left(-9.07398 \left(c^2-1\right) x^6+1.51233 \left(3 c^2+1\right) x^6-1.31754 \left(3 (3 x-4) c^2-21 x+20\right) x^4+1.31754 \left(2 \left(L^2+6 x-6\right) c^2-L^2-3.44353 \left(c^2-1\right) x^2+6 x-4\right) x^4+4.59137 (x-1) (5 x-4) x^2+0.573921 \left(\left(8 (x-1) L^2+3 (x (9 x-16)+8)\right) c^2-20.6612 \left(c^2-1\right) (x-1) x^2-8 L^2 (x-1)+3 x (5 x-8)+8\right) x^2-4 (x-1)^2 \left(L^2-1.14784 x^2\right)\right) \chi ^{(2,0)}(x,t) x^4}{0.573921 x^2+x-1}-8.69577 \sqrt{0.573921 x^2+x-1} \chi ^{(2,1)}(x,t) x^4-\frac{8.69577 \left((1.14784 x+1) \chi ^{(1,1)}(x,t)+2 \left(0.573921 x^2+x-1\right) \chi ^{(2,1)}(x,t)\right) x^4}{\sqrt{0.573921 x^2+x-1}}-34.7831 \sqrt{0.573921 x^2+x-1} \chi ^{(1,1)}(x,t) x^3-\frac{0.757576 \left(-1.31509 \left(3 c^2+1\right) x^9-15.781 x^9-1.1457 \left(\left(2 L^2+21 x-18\right) c^2-L^2-24.1047 \left(c^2-1\right) x^2+10 x-6\right) x^7-0.756165 \left(3.0303 \left(\left(2 L^2+9 x-6\right) c^2-L^2+37 x-32\right)-10.4349 \left(c^2-1\right) x^2\right) x^7-0.998137 \left(\left(8 (x-1) L^2+51 x^2-84 x+36\right) c^2+L^2 (6-7 x)+x (27 x-40)+3.44353 x^2 \left((30-29 x) c^2+35 x-34\right)+12\right) x^5-0.658771 \left(-46.9572 \left(c^2-1\right) (x-1) x^2-0.757576 \left(-\left(32 (x-1) L^2+117 x^2-216 x+96\right) c^2+20 L^2 (x-1)-x (313 x-536)-224\right)\right) x^5+2.29568 (x-1) \left(15.6524 \left(c^2-1\right) (x-1) x^2-0.757576 \left(\left(8 (x-1) L^2+3 (x (9 x-16)+8)\right) c^2-8 L^2 (x-1)+11 x (5 x-8)+32\right)\right) x^3-0.434789 \left(\left(16 L^2 (x-1)^2+3 (x (x (27 x-68)+56)-16)\right) c^2-8 L^2 (x-1) (4 x-3)+2.29568 (x-1) x^2 \left((96-81 x) c^2+163 x-152\right)+x (9 x (5 x-12)+80)-16\right) x^3+6.06061 (x-1)^3 \left(L^2-1.14784 x^2\right) x+3.0303 (x-1)^2 \left((3 x-2) L^2+4.59137 (4-5 x) x^2\right) x\right) \chi ^{(1,0)}(x,t) x^2}{\left(0.573921 x^2+x-1\right)^2}+2.29568 \chi ^{(2,2)}(x,t) x^2+\frac{0.757576 L^2 \left(1.99627 (5 c-1) x^5-1.73915 \left(c \left(L^2+2 x-2\right)-1.72176 (c-1) x^2\right) x^3+1.73915 (-5 x+c (9 x-10)+4) x^3-3.0303 (x-1) (3 x-2) x-0.757576 \left(c \left(4 (x-1) L^2+x (9 x-16)+8\right)-2.29568 (4 c-3) (x-1) x^2\right) x\right) \chi (x,t) x}{0.573921 x^2+x-1}+\frac{1.51515 L^2 (1.14784 x+1) \chi ^{(0,1)}(x,t) x}{\sqrt{0.573921 x^2+x-1}}+6.06061 L^2 \sqrt{0.573921 x^2+x-1} \chi ^{(1,1)}(x,t) x+4.59137 \chi ^{(1,2)}(x,t) x+3.0303 L^2 \sqrt{0.573921 x^2+x-1} \chi ^{(0,1)}(x,t)-\frac{4 L^2 \chi ^{(0,2)}(x,t)}{x^2})=0$